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$: INNOVATIVE ASPECTS OF RESEARCH … › gtusitecirculars › ...It is certi ed that PhD thesis entitled \Innovative aspects of research advance-ment in graph theory" by Adalja Divya$

INNOVATIVE ASPECTS OF RESEARCH

ADVANCEMENT IN GRAPH THEORY

A Thesis submitted to Gujarat Technological University

for the Award of

Doctor of Philosophy

in

Science-Maths

by

Adalja Divya Ghanshyambhai

(Enrollment No.: 149997673001)

under supervision of

Dr. Gaurang V. Ghodasara

GUJARAT TECHNOLOGICAL UNIVERSITY

AHMEDABAD

FEBRUARY 2020

c©Adalja Divya Ghanshyambhai

i

DECLARATION

I declare that the thesis entitled “Innovative aspects of research advancement

in graph theory” submitted by me for the degree of Doctor of Philosophy is the

record of research work carried out by me during the period from May 2015 to

September 2019 under the supervision of Dr. Gaurang V. Ghodasara and this

has not formed the basis for the award of any degree, diploma, associateship, fel-

lowship, titles in this or any other University or other institution of higher learning.

I further declare that the material obtained from other sources has been duly ac-

knowledged in the thesis. I shall be solely responsible for any plagiarism or other

irregularities, if noticed in the thesis.

Signature of Research Scholar : Date : 20/02/2020

Name of Research Scholar : Adalja Divya Ghanshyambhai

Enrollment No. : 149997673001

Place : Rajkot

ii

CERTIFICATE

I certify that the work incorporated in the thesis entitled “Innovative aspects

of research advancement in graph theory” submitted by Ms. Adalja Divya

Ghanshyambhai was carried out by the candidate under my supervision/guidance.

To the best of my knowledge:

(i) the candidate has not submitted the same research work to any other institution

for any Degree/Diploma, Associateship, Fellowship or other similar titles.

(ii) the thesis submitted is a record of original research work done by the Research

Scholar during the period of study under my supervision, and

(iii) the thesis represents independent research work on the part of the Research

Scholar.

Signature of Supervisor : Date : 20/02/2020

Name of Supervisor : Dr. Gaurang V. Ghodasara

Assistant Professor in Mathematics

H. & H. B. Kotak Institute of Science, Rajkot- 360001.

Place : Rajkot

iii

Course-work Completion Certificate

This is to certify that Ms. Adalja Divya Ghanshyambhai, Enrolment number:

149997673001, is a PhD scholar enrolled for PhD program in the branch Science

- Maths of Gujarat Technological University, Ahmedabad.

(Please tick the relevant option(s))

� She has been exempted from the course-work (successfully completed during

M.Phil. Course).

� She has been exempted from Research Methodology Course only (successfully

completed during M.Phil. Course)

� She has successfully completed the PhD course work for the partial requirement

for the award of PhD Degree. Her performance in the course work is as follows:

Grade Obtained in Research Method-

ology (PH001)

Grade Obtained in Self Study Course

(Core Subject) (PH002)

BC AB

Signature of Supervisor :


iv

Originality Report Certificate

It is certified that PhD thesis entitled “Innovative aspects of research advance-

ment in graph theory” by Adalja Divya Ghanshyambhai has been examined by

us. We undertake the following:

(a) Thesis has significant new work/knowledge as compared to already published

or are under consideration to be published elsewhere. No sentence, equation,

diagram, table, paragraph or section has been copied verbatim from previous

work unless it is placed under quotation marks and duly referenced.

(b) The work presented is original and own work of the author (i.e. there is no

plagiarism). No ideas, processes, results or words of others have been presented

as Author’s own work.

(c) There is no fabrication of data or results which have been compiled/analyzed.

(d) There is no falsification by manipulating research materials, equipment or pro-

cesses, or changing or omitting data or results such that the research is not

accurately represented in the research record.

(e) The thesis has been checked using Turnitin (copy of originality report at-

tached) and found within limits as per GTU Plagiarism Policy and instructions

issued from time to time (i.e. permitted similarity index < 10 %).




Place : Rajkot



Place : Rajkot

v

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PhD THESIS Non-Exclusive License to

GUJARAT TECHNOLOGICAL UNIVERSITY

In consideration of being a PhD Research Scholar at GTU and in the interests of the

facilitation of research at GTU and elsewhere, I, Adalja Divya Ghanshyambhai

having Enrollment Number 149997673001 hereby grant a non-exclusive, royalty

free and perpetual license to GTU on the following terms:

(a) GTU is permitted to archive, reproduce and distribute my thesis, in whole or

in part, and/or my abstract, in whole or in part (referred to collectively as the

“Work”) anywhere in the world, for non-commercial purposes, in all forms of

media.

(b) GTU is permitted to authorize, sub-lease, sub-contract or procure any of the

acts mentioned in paragraph (a).

(c) GTU is authorized to submit the Work at any National/International Library,

under the authority of their “Thesis Non-Exclusive License.”

(d) The Universal Copyright Notice c© shall appear on all copies made under the

authority of this license.

(e) I undertake to submit my thesis, through my University, to any Library and

Archives. Any abstract submitted with the thesis will be considered to form

part of the thesis.

(f) I represent that my thesis is my original work, it does not infringe any rights

of others, including privacy rights, and that I have the right to make the grant

conferred by this non-exclusive license.

(g) If third party copyrighted material was included in my thesis for which, under

the terms of the Copyright Act, written permission from the copyright owners is

required, I have obtained such permission from the copyright owners to do the

acts mentioned in paragraph (a) above for the full term of copyright protection.

(h) I retain copyright ownership and moral rights in my thesis, and may deal with

the copyright in my thesis, in any way consistent with rights granted by me to

vii

my University in this non-exclusive license.

(i) I further promise to inform any person to whom I may hereafter assign or

license my copyright in my thesis of the rights granted by me to my University

in this non-exclusive license.

(j) I am aware of and agree to accept the conditions and regulations of PhD in-

cluding all policy matters related to authorship and plagiarism.




Place : Rajkot



Assistant Professor in Mathematics

H. & H. B. Kotak Institute of Science, Rajkot- 360001.

Place : Rajkot

viii

Thesis Approval Form

The viva-voce of the PhD Thesis submitted by Ms. Divya Ghanshyambhai Adalja

(Enrollment No. 149997673001) entitled “Innovative aspects of research ad-

vancement in graph theory” was conducted on

(day and date) at Gujarat Technological University.

(Please tick any one of the following option.)

� The performance of the candidate was satisfactory. We recommend that he/she

shall be awarded the PhD degree.

� Any further modifications in research work recommended by the panel after 3

months from the date of first viva-voce upon request of the Supervisor or request

of Independent Research Scholar after which viva-voce can be re-conducted by the

same panel again.

(briefly specify the modifications suggested by the panel)

� The performance of the candidate was unsatisfactory. We recommend that he/she

should not be awarded the PhD degree.

(The panel must give justifications for rejecting the research work)

Signature of Supervisor with seal :


(External Examiner 1) Name and Signature :



ix

ABSTRACT

The theory of graphs mainly evolved with the rise of the computer age. It is one

such field of mathematics with cuts across a wide range of disciplines of human

understanding. It has rigorous applications in diversified fields such as computer

science, social sciences, engineering, physics, chemistry and biology. Graphs have

been proved to be a powerful mathematical tool to explain structures of molecules

and it is also possible to explain flow of control with the help of a graph structure.

Development of computer science boost up the research work in this field. There

are many interesting fields of research in graph theory. Decomposition of graphs,

Domination number of graphs, Chromatic graph theory, Theory of hypergraph, Al-

gebraic graph theory, Labeling of graphs and Enumeration of graphs are several

branches of research work in graph theory in various directions.

The field of graph theory has become a field of multifaceted applications ranging

from neural network to bio-technology and coding theory to mention a few. Graphs

are very much useful to solve many problems which are complex in nature but seem-

ingly understandable. The Konigsberg Bridge problem, Four Color problem, Around

the World Game and Traveling Salesman problem are few examples of this character.

Graph labeling is an assignment of numbers/values to vertices or edges or both.

The labeling of graphs is one of the emerging areas of research due to its varied

x

applications. The problems related to labeling of graphs challenge to our mind for

their eventual solutions. In this thesis, we have mainly focused upon the graph

families which satisfy the conditions of different graph labeling techniques such as

divisor cordial labeling, square divisor cordial labeling, cube divisor cordial label-

ing, vertex odd divisor cordial labeling and sum divisor cordial labeling of graphs.

Throughout the thesis, we have considered simple, finite, undirected and connected

graph G = (V,E) with order p and size q.

The present work contains a bonafied record of the research work carried out on

the concepts of graph labeling and contains report of investigations concern to the

concepts of graph labeling.

xi

Acknowledgement

First I sincerely thank The Almighty for the grace showered on me in complet-

ing this research work.

I would like to express my sincere gratitude to my research supervisor and men-

tor Dr. G. V. Ghodasara, Assistant Professor, H. & H. B. Kotak Institute of

Science, Rajkot for his patience, motivation, continuous support, encouragement,

wide experience and immense knowledge to make my research work. His guidance

helped me during the time of research and writing of this thesis.

My special gratitude goes to the Doctorate Progress Committee (DPC) members:

Dr. N. H. Shah, Lecturer, Government Polytechnic, Jamnagar and Dr. N. A.

dani, Senior Lecturer, Government Polytechnic, Rajkot for their precious presence

at every Doctoral Progress Committee (DPC) and providing valuable suggestions

which ensured that the research becomes more significant.

During course of my journey, I have referred many books and good number of

research papers on related topics. I am thankful to the concerned authors.

I would like to express my deep sense of gratitude to Dr R. B. Jadeja, Dr. R.

L. Jhala, the management and the staff members of Marwadi Engineering college

for their sincere support and inspiration during my research journey.

Thanks to my coresearchers Prof. Mitesh Patel and Prof. Mohit Bosmia

for extraordinary help, fruitful suggestions and moral support wherever required.

How can I forget my beloved and respected teachers of school days? The interest

in mathematics grew due to their efforts and expert teaching. I extend deep sense

of gratitude towards them.

I owe my huge debt of thanks to my parents Ghanshyambhai Adalja and

Hiraben Adalja as well as my husband Pinank Patel for their moral support,

encouragement and motivation. Whatever I have achieved in my life is a result of

the blessings and sacrifice of my parents and my husband.

At last I convey my sincere thanks to all those who have provided their kind

support but I have missed to mention them.

Adalja Divya Ghanshyambhai

xii

Contents

List of Nomenclatures xvi

List of Figures xvii

1 Introduction 1

1.1 Definition of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objective and scope of work . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Original contribution by the thesis . . . . . . . . . . . . . . . . . . . 3

1.4 Methodology of research . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Achievements with respect to objectives . . . . . . . . . . . . . . . . 5

1.6 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Review of Literature 8

2.1 Historical information of graph theory . . . . . . . . . . . . . . . . . 8

2.2 Basic terminologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Basic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Operations on a graph / Operations of graphs . . . . . . . . . 11

2.3 Graph labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Divisor Cordial Labeling With The Use of Some Graph Operations 17

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Some Known Results on DC Labeling . . . . . . . . . . . . . . . . . . 18

3.3 Some New DC Graphs With the Use of Ringsum Operation . . . . . 20

xiii

CONTENTS

3.4 DC Labeling With the Use of Switching Invariance in Cycle Allied

Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5 DC Labeling With the Use of Duplication of a Vertex/Edge in Star

Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6 Conclusion and Scope for Further Research . . . . . . . . . . . . . . . 44

4 Square Divisor Cordial, Cube Divisor Cordial and Vertex Odd Di-

visor Cordial Labeling of Graphs 45

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 Square DC Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.2 Some Known Results on Square DC Labeling . . . . . . . . . 46

4.3 Cube DC Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3.2 Some Known Results on Cube DC Labeling . . . . . . . . . . 49

4.4 Vertex Odd DC Labeling . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4.2 Some Known Results on VODC Labeling . . . . . . . . . . . . 50

4.5 New Results on Square DC, Cube DC and Vertex Odd DC Labeling

of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.6 VODC Labeling With the Use of Switching of a Vertex in Cycle Allied

Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.7 VODC Labeling With the Use of Switching of a Vertex in Wheel and

Shell Allied Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.8 VODC Labeling With the Use of Ringsum of Different Graphs with

Star Graph K1,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77


5 Sum Divisor Cordial Labeling of Graphs 89

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2 Some Existing Results on Sum DC Labeling . . . . . . . . . . . . . . 90

xiv

CONTENTS

5.3 Some New Cycle Related Sum DC Graphs . . . . . . . . . . . . . . . 91

5.4 SDC Labeling of Snakes Related Graphs . . . . . . . . . . . . . . . . 103


6 Sum Divisor Cordial Labeling With the Use of Some Graph Oper-

ations 118

6.1 SDC Labeling of Graphs With the Use of Ringsum of Different Graphs

with Star Graph K1,n . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.2 SDC Labeling in the Graphs constructed from Corona Product with

K1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.3 SDC Labeling With the Use of Switching of a Vertex in Cycle Allied

Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.4 SDC Labeling With the Use of Switching of a Vertex in Wheel and

Shell Allied Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.5 SDC Labeling by Duplicating a Vertex/Edge in Star Graph . . . . . . 183

6.6 SDC Labeling by Duplicating Vertex/Edge in Cycle Graph . . . . . . 188

6.7 SDC Labeling by duplicating Vertex/Edge in Path Graph . . . . . . . 196


7 Summary 205

References 208

Annexure 212

xv

CONTENTS

List of Nomenclatures

Nomenclature Meaning

V (G) Vertex set of a graph G

E(G) Edge set of a graph G

|B| Cardinality of set B

d(v) or dG(v) Degree of a vertex v in a graph G

Pn Path graph with n vertices

Cn Cycle graph with n vertices

Kn Complete graph with n vertices

Km,n Complete bipartite graph with m+ n vertices

K1,n Star graph with n+ 1 vertices

Wn Wheel graph with n+ 1 vertices

Gn Gear graph with 2n+ 1 vertices

Sn Shell graph with n vertices

Hn Helm graph with 2n+ 1 vertices

CHn Closed helm graph with 2n+ 1 vertices

Fn Fan graph with n+ 1 vertices

DFn Fan graph with n+ 2 vertices

Fln Flower graph with 2n+ 1 vertices

C(k)n One point union of k copies of cycle Cn

Cn,3 Cycle with twin chords

Cn(1, 1, n− 5) Cycle with triangle

Bn,n Bistar graph with 2n+ 2 vertices

U(m,n) Umbrella graph with m+ n vertices

xvi

CONTENTS

Nomenclature Meaning

Tn Triangular snake

Qn quadrilateral snake

DTn Double triangular snake

DQn Double quadrilateral snake

A(Tn) Alternate triangular snake

A(Qn) Alternate quadrilateral snake

DA(Tn) Double alternate triangular snake

DA(Qn) Double alternate quadrilateral snake

G⊕H Ringsum of two graphs G and H

G⊙

H Corona of two graphs G and H

Pn⊙

K1 Comb graph with 2n vertices

Cn⊙

K1 Crown graph with 2n vertices

ACn Armed crown graph with 3n vertices

G ∪H Union of two graphs G and H

S ′(G) Splitting graph of a graph G

DS(G) Degree splitting graph of a graph G

D2(G) Shadow graph of a graph G

G2 Square of a graph G

N(v) Neighbourhood of vertex v

dne Least integer not less than real number n (Ceiling of n)

bnc Greatest integer not greater than real number n (Floor of n)

DC Divisor Cordial

CDC Cube Divisor Cordial

VODC Vertex Odd Divisor Cordial

SDC Sum Divisor Cordial

W.L.O.G. Without loss of generality

xvii

List of Figures

3.1 DC labeling in W5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 DC labeling in C5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 DC labeling in the graph constructed from ringsum of C6 with one

chord and K1,6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 DC labeling in C7,3 ⊕K1,7 . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 DC labeling in C8(1, 1, 3)⊕K1,8 . . . . . . . . . . . . . . . . . . . . . 25

3.6 DC labeling in W5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.7 DC labeling in Fl4 ⊕K1,4 . . . . . . . . . . . . . . . . . . . . . . . . 28

3.8 DC labeling in S7 ⊕K1,7 . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.9 DC labeling in P5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.10 DC labeling in DF5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . . . 32

3.11 DC labeling in K2,7 ⊕K1,7 . . . . . . . . . . . . . . . . . . . . . . . . 33

3.12 DC labeling in (G6)v . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.13 DC labeling in (S7)v . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.14 DC labeling in (Fl4)v . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.15 DC labeling in the graph constructed from duplication of vertex by

edge in K1,5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.16 DC labeling in the graph constructed from duplication of edge v0v8

in K1,8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1 Square DC labeling in S7 . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 CDC labeling in K2,7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3 VODC labeling in Fl7 . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4 Square DC labeling in K1,1,6 . . . . . . . . . . . . . . . . . . . . . . . 53

xviii

LIST OF FIGURES

4.5 Square DC labeling in U(9, 3) . . . . . . . . . . . . . . . . . . . . . . 55

4.6 Square DC labeling in C(5)4 . . . . . . . . . . . . . . . . . . . . . . . . 56

4.7 Square DC labeling in < K(1)1,5 , K

(2)1,5 > . . . . . . . . . . . . . . . . . . 57

4.8 Square DC labeling in arbitrary supersubdivision of K1,4 . . . . . . . 58

4.9 Square DC labeling in the graph constructed from duplication of an

edge in K1,8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.10 Square DC labeling in K2,5 � u2(K1) . . . . . . . . . . . . . . . . . . 61

4.11 VODC labeling in (G)v. . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.12 VODC labeling in (C8,3)v . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.13 VODC labeling in (C8(1, 1, 3))v . . . . . . . . . . . . . . . . . . . . . 67

4.14 VODC labeling in (W9)v1 . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.15 VODC labeling in (G6)v . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.16 VODC labeling in (S7)v . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.17 VODC labeling in (Fl4)v . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.18 VODC labeling in the graph constructed from ringsum of C7 with

one chord and K1,7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.19 VODC labeling in C8,3 ⊕K1,8 . . . . . . . . . . . . . . . . . . . . . . 79

4.20 VODC labeling in C8(1, 1, 3)⊕K1,8 . . . . . . . . . . . . . . . . . . . 81

4.21 VODC labeling in P5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . . 82

4.22 VODC labeling in W6 ⊕K1,6 . . . . . . . . . . . . . . . . . . . . . . . 83

4.23 VODC labeling in Fl4 ⊕K1,4 . . . . . . . . . . . . . . . . . . . . . . 84

4.24 VODC labeling in K2,7 ⊕K1,7 . . . . . . . . . . . . . . . . . . . . . . 86

4.25 VODC labeling in DF5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . 87

5.1 SDC labeling in K1,7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2 SDC labeling in C5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3 SDC labeling in C6 with one chord . . . . . . . . . . . . . . . . . . . 93

5.4 SDC labeling in C7,3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.5 SDC labeling in C8(1, 1, 3) . . . . . . . . . . . . . . . . . . . . . . . . 95

5.6 SDC labeling in W5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.7 SDC labeling in H6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.8 SDC labeling in Wb5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

xix

LIST OF FIGURES

5.9 SDC labeling in S7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.10 SDC labeling in Fl4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.11 SDC labeling in DF5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.12 SDC labeling in T6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.13 SDC labeling in DT5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.14 SDC labeling in Q5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.15 SDC labeling in DQ5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.16 SDC labeling in A(T7) . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.17 SDC labeling in A(Q8) . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.18 SDC labeling in DA(T10) . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.19 SDC labeling in DA(Q9) . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.1 SDC labeling in C5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . . . 120

6.2 SDC labeling in the graph constructed from ringsum of C7 with one

chord and K1,7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.3 SDC labeling in C7,3 ⊕K1,7 . . . . . . . . . . . . . . . . . . . . . . . 122

6.4 SDC labeling in C8(1, 1, 3)⊕K1,8 . . . . . . . . . . . . . . . . . . . . 123

6.5 SDC labeling in W6 ⊕K1,6 . . . . . . . . . . . . . . . . . . . . . . . . 124

6.6 SDC labeling in Fl4 ⊕K1,4 . . . . . . . . . . . . . . . . . . . . . . . . 126

6.7 SDC labeling in G6 ⊕K1,6 . . . . . . . . . . . . . . . . . . . . . . . . 127

6.8 SDC labeling in P5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . . . 128

6.9 DC labeling in S7 ⊕K1,7 . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.10 SDC labeling in DF5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . . 131

6.11 SDC labeling in K2,7 ⊕K1,7 . . . . . . . . . . . . . . . . . . . . . . . 132

6.12 SDC labeling in K1,6 �K1 . . . . . . . . . . . . . . . . . . . . . . . . 134



6.15 SDC labeling in W7 �K1 . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.16 SDC labeling in H7 �K1 . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.17 SDC labeling in Fl7 �K1 . . . . . . . . . . . . . . . . . . . . . . . . 143

6.18 SDC labeling in F8 �K1 . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.19 SDC labeling in DF6 �K1 . . . . . . . . . . . . . . . . . . . . . . . . 147

xx

LIST OF FIGURES

6.20 SDC labeling in S(K1,5)�K1,5 . . . . . . . . . . . . . . . . . . . . . 148

6.21 SDC labeling in corona of C6 with one chord and K1 . . . . . . . . . 149

6.22 SDC labeling in C7,3 �K1 . . . . . . . . . . . . . . . . . . . . . . . . 150

6.23 SDC labeling in C8(1, 1, 3)�K1 . . . . . . . . . . . . . . . . . . . . . 151

6.24 SDC labeling in (G)v. . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.25 SDC labeling in (C8,3)v. . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.26 SDC labeling in (C7(1, 1, 2))v . . . . . . . . . . . . . . . . . . . . . . 159

6.27 SDC labeling in (W9)v1 . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.28 SDC labeling in (G6)v. . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.29 SDC labeling in (S7)v . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.30 SDC labeling in (H6)v. . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.31 SDC labeling in (CH6)v . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.32 SDC labeling in (Fl4)v . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.33 SDC labeling in (B4,5)v1 . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.34 SDC labeling in (B4,5)u0 . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.35 SDC labeling in (B4,5)v0 . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.36 SDC labeling in (P5 �K1)v1 . . . . . . . . . . . . . . . . . . . . . . . 176

6.37 SDC labeling in (P5 �K1)u1 . . . . . . . . . . . . . . . . . . . . . . . 176

6.38 SDC labeling in (P5 �K1)u2 . . . . . . . . . . . . . . . . . . . . . . . 177

6.39 SDC labeling in (C7 �K1)v1 . . . . . . . . . . . . . . . . . . . . . . . 178

6.40 SDC labeling in (C7 �K1)u1 . . . . . . . . . . . . . . . . . . . . . . . 178

6.41 SDC labeling in (AC5)v1 . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.42 SDC labeling in (AC5)w1 . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.43 SDC labeling in (AC5)u1 . . . . . . . . . . . . . . . . . . . . . . . . . 183

6.44 SDC labeling in the graph constructed by duplicating edge in K1,8 . . 184

6.45 SDC labeling in the graph constructed by duplicating apex vertex v0

by edge v′0v′′0 in K1,5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

6.46 SDC labeling in the graph constructed by duplicating vertex v7 by

edge v′7v′′7 in K1,7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

6.47 SDC labeling in the graph constructed by duplicating an edge by a

vertex in K1,6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

xxi

LIST OF FIGURES

6.48 SDC labeling in the graph constructed by duplicating a vertex in C5 . 189

6.49 SDC labeling in the graph constructed by duplicating an edge in C6 . 191

6.50 SDC labeling in the graph constructed by duplicating vertex by edge

in C5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

6.51 SDC labeling in the graph constructed by duplicating each vertex by

new edge in cycle C5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6.52 SDC labeling in the graph constructed by duplicating edge by new

vertex in C7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

6.53 SDC labeling in the graph constructed by duplicating a vertex in P5 . 197

6.54 SDC labeling in the graph constructed by duplicating edge in P5 . . . 199

6.55 SDC labeling in the graph constructed by duplicating vertex v′2 by a

new edge v′2v′′2 in P5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

6.56 SDC labeling in the graph constructed by duplicating an edge v2v3

by a new vertex v′ in P5 . . . . . . . . . . . . . . . . . . . . . . . . . 201

6.57 SDC labeling in the graph constructed by duplicating each vertex by

edge in P5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

xxii

CHAPTER 1

Introduction

This chapter is of introductory nature which gives glimpse of the work embodied

in the thesis and its aim is to provide fundamental components of research. The

essence of results constructed in the other chapters is summarized in this chapter.

1.1 Definition of the problem

The development in the subject of graph labeling is due to painstacking efforts of

many researchers in this field towards solving “Ringel-Kotzig conjecture”. This con-

jecture is considered to be one of the root cause of development in several labeling

techniques like harmonious labeling, cordial labeling, k-equitable labeling etc.

From obtainable written as well as eletronic type literature sources on completely

different topics of graph labeling, some fascinating facts and unsolved problems are

found.

Divisors of a natural number n are the numbers, when divided by n, leaves remainder

0. For example, prime divisors of 210 are 2, 3, 5, 7.

Combining the concepts of divisor of a number from number theory and cordial

labeling from graph labeling, Varatharajan[44] introduced one of the variant of cor-

dial labeling namely DC labeling. Many research papers have been published in this

topic and hence several DC graphs are found. For any two natural numbers a and

b, instead of considering a | b or b | a, if one consider a2 | b or a3 | b then these

will give rise to square DC/cube DC labeling. Some additional changes within the

condition of DC labeling give rise to sum DC labeling and vertex odd DC labeling.

1

1.2. Objective and scope of work

It is interesting to see whether a certain graph which admit one of these labeling

will admit other or not. This thought may produce certain relation/condition be-

tween these invariants of DC labeling. We have focused upon deriving graph families

satisying/not satisfying these invariants of DC labeling.

If a graph satisfies a particular invariant of DC labeling, then “applying different

graph operations, shall the graph preserve that labeling or not” will also be intrest-

ing to see. Here we have considered various graph operations on a certain graph

family and derived the conclusion whether it admits a particular graph labeling or

not. In some cases, we have concluded that some graph operations are labeling

preserving.

In this thesis we have mainly focussed on DC labeling, square DC labeling, cube

DC labeling, vertex odd DC labeling and sum DC labeling. We have considered the

graph operations such as ringsum of different graphs, switching of a vertex, dupli-

cation of vertex/edge, corona product and arbitrary super subdivision.

1.2 Objective and scope of work

The main objective of present research work is to generate new direction to gain

knowledge in the area of graph labeling. Graph labeling is aimed to cover a diversity

of applications in manifold fields. After studying different graph labeling techniques,

the following objectives may be fulfilled.

z Some new graph labeling techniques using/combining the concepts of number

theory, combinatorics and graph theory can be constructed.

z Different graph labelings for the graphs constructed from graph operations can

be investigated.

z Results related to investigations with the use of other graph labeling techniques

and for various graph families can be carried out.

z Using combination of theoretical knowledge and independent mathematical

thinking, one can explore the graphs which satisfy particular graph labeling

technique.

2

1.3. Original contribution by the thesis

z Some new graph labeling techniques may be invented which will give new direc-

tion to young researchers for the development in the field of research in graph

labeling.

z One can identify and explore the family of graphs which satisfy certain graph

labeling techniques but not the other one with particular reason. It helps to

relate different labeling techniques.

z Intensive study of the results derived in this thesis may help to solve conjectures

and open problems.

There is a good scope to investigate equivalent results for different graph families.

Few such problems are stated below.

z Derive essential and adequate condition (if possible) for a graph to become a

DC graph.

z Derive results on DC labeling for generalized petersen graph P (n, k).

z Derive results on DC labeling for line graph, middle graph and total graph of

different graph families.

z Derive results on DC labeling for snake related graphs.

z Derive results on square and cube DC labeling for different operations of two

graphs such as union, ringsum, corona etc.

z Derive results on vertex odd DC labeling for complete lattice grid and related

graphs.

z Derive new graphs on sum DC labeling for line graph, middle graph and total

graph of different graph families.

1.3 Original contribution by the thesis

In the presented research work, different results on DC labeling and its variants

such as square DC labeling, cube DC labeling, vertex odd DC labeling and sum DC

labeling are derived.

DC labeling of graphs:

3

1.3. Original contribution by the thesis

z We have studied some properties of DC graph and we discussed some new DC

graphs by using ringsum operation.

z We have also discussed DC labeling by using duplication of vertex/edge of

special graphs.

z We have established that the graph constructed from switching invariance in

some graph families admit a DC labeling.

Square DC labeling of graphs:

z We have proved that K1,1,n, Um,n, one point union of t copies of the cycle C4,

< K(1)1,n, K

(2)1,n >, arbitrary super subdivision of K1,n, duplication of an edge in

K1,n, K2,n � u2(K1) are square DC graphs.

Cube DC labeling of graphs:

z We have established the results in cube DC labeling similar to the results

established for square DC labeling.

Vertex odd DC labeling of graphs:

z We have studied some properties of vertex odd DC graph and we discuss vertex

odd DC labeling by using ringsum of graphs.

z We also have derived the results of vertex odd DC labeling similar to the results

derived for square DC labeling.

z We have established that the graph constructed from switch of a vertex in some

specific graph families admit a vertex odd DC labeling.

Sum DC labeling of graphs:

z We have proved that cycle, wheel, shell, fan related graphs are sum DC.

z We also have derived that snakes related graphs are sum DC.

z We have discussed sum DC labeling with the use of ringsum and corona product

of two graphs.

z We have studied sum DC labeling by using duplication of vertex/edge in some

graphs.

4

1.4. Methodology of research

z We have established that some graphs constructed from switching invariance

in certain graph families admit sum DC labeling.

1.4 Methodology of research

Initially I started with reading books and research papers based on graph label-

ing which are published in reputed journals. This helped me to develop cohesive

and conceptual thinking. I have adopted some mathematical concepts related to

graph theory and number theory from available sources of literature in printed and

electronic form. Parallelly I focused on LATEX, which is an effective tool for high-

quality typesetting for publication of research work. Later was the study of research

methodology tools such as expansion of past work, modification of mathematical

results, development of new results and efforts to solve conjecture used for further

research work. By refering e-content on combinatorics, I have got proper direction

to work on different graph labeling techniques which is found to be combination

of combinatorics, number theory and graph theory. The concept of divisor of a

number leads to the labeling techniques such as DC labeling, square DC labeling,

cube DC labeling, vertex odd DC labeling and sum DC labeling. Combinatorial and

induction methods have also been used to construct and verify labeling pattern for

a particular graph lebeling defined for the given graph family.

1.5 Achievements with respect to objectives

Since the registration of Ph.D., nine research papers have been published in referred

international journals and one research paper is presented in national conference.

Study of various existing graph labeling techniques for different graph families is a

part of literature survey. We have investigated different graph families which satisfy

DC labeling and main four invariants of it.

1.6 Thesis organization

The thesis consists of seven chapters which delivers the content including some de-

rived results and open problems in different graph labeling techniques.

5

1.6. Thesis organization

Chapter 1 is of introductory nature which gives glimpse of the work embodied

in the thesis and its aim is to provide fundamental components of research in the

present work.

Chapter 2 is basically intended to provide historical information and broad concept

of graph theory as well as essential terminology required for the pertinent work.

Chapter 3 describes DC labeling for some results constructed from different graph

operations such as ringsum of different graphs with star graph, vertex switching and

duplication of vertex.

Chapter 4 aims to discuss square DC, cube DC and vertex odd DC labeling of

graphs. We originate square DC, cube DC and vertex odd DC labeling for some ba-

sic graphs like K1,1,n, one point union C(t)4 of t copies of cycle C4, umbrella U(m,n)

(m,n > 2) etc. We also derive that the graph constructed from arbitrary super

subdivision of K1,n and graph constructed from duplication of an edge in K1,n are

square DC, cube DC and vertex odd DC graphs. We have studied some properties

of vertex odd DC graph and we discuss vertex odd DC labeling for the graphs con-

structed from graph operations ringsum of graphs. We have established that the

graphs constructed from switching of a vertex in some specific graph families admit

vertex odd DC labeling.

Chapter 5 deals with sum DC labeling of graphs. We have proved that some basic

graphs like cycle, wheel, helm, shell, double fan, flower and web are sum DC graphs.

We have also constructed some new snake related sum DC graphs.

Chapter 6 describes sum DC labeling for the graphs constructed from graph oper-

ations such as ringsum of different graphs with star graph K1,n, corona of different

graphs with graph K1, vertex switching of graphs, duplication of a vertex in star,

cycle and path related graphs.

Chapter 7 contains summary of previous chapters. It also includes details of re-

search publications based on the present research work.

Furthermore, at the end of each chapter some open problems based on the con-

templation of results of the current study are presented. These open problems are

6

1.6. Thesis organization

presented with a view to be helpful to young researchers in the field of graph theory.

The references used throughout this work are listed at the end.

7

CHAPTER 2

Review of Literature

This chapter is basically intended to provide historical information and broad idea

of graph theory as well as essential terminology required for the pertinent work.

2.1 Historical information of graph theory

The history of graph theory may be specifically traced from 1735, when Leonhard

Euler (Swiss Mathematician) solved the Konigsberg bridge problem. Euler repre-

sented the first paper in 1736 entitled Solution of a Problem Relating to the Geometry

of Position, which is supposed to be the birth of graph theory. First book on graphs

and related literature was written by Denes Konig in 1936. Another book entitled

Graph Theory was written by Frank Harary in 1969. It was considered the world

over to be the definitive textbook on the subject.

Cayley used graph theory for the study of particular analytical forms. One of the

most famous problems in graph theory is the four color problem which states that,

Is it true that any map drawn in the plane may have regions colored with four colors,

in such a way that any two regions having common border have different colors ?

This problem was first posed by Francis Guthrie in 1852 and first written record of

this problem is in a letter of De Morgan addressed to Hamilton in the same year.

This well celebrated problem took hundred years for its solution. In 1976, Walfgang

Haken and Kenneth Appel solved this problem by giving very lengthy proof.

8

2.2. Basic terminologies

2.2 Basic terminologies

Let G = (V,E) be a graph consists of two finite sets; a non empty set of vertices

V (G) and set of edges E(G) (may or may not be empty). The members of V (G)

and E(G) are commonly termed as graph elements. The cardinality of the vertex

set of a graph G is called order of G whereas the cardinality of its edge set is called

size of G.

Throughout this thesis, we consider a graph G to be simple, undirected, finite and

connected. If |V (G)| = p and |E(G)| = q then we write it as G = (p, q) graph.

2.2.1 Basic graphs

A graph without loops and multiple edges is called a simple graph. The degree

of a vertex v (deg(v) or d(v)) of a graph G is the number of edges incident to

the vertex, with loops counted twice. It is denoted by deg(v) or d(v). A pendant

vertex is a vertex of degree one. A walk in a graph G is a finite sequence W =

v0e1v1e2v2, . . . , vk−1ekvk whose terms are alternately vertices and edges such that,

for 1 ≤ i ≤ k, the edge ei has ends vi−1 and vi. The length of a walk is the number

of edges in it. Path Pn is special walk in which vertex repeatation is not allowed. A

closed path is called a cycle Cn.

Definition 2.2.1 (Bondy and Murty[16]). Wheel Wn is the graph constructed by

join of the graphs Cn and K1. i.e. Wn = Cn +K1. Here the vertices corresponding

to Cn are called rim vertices and Cn is called rim of Wn, while the vertex corresponds

to K1 is called apex vertex.

Definition 2.2.2 (Ma and Feng[22]). Gear Gn is the graph constructed from wheel

Wn by subdividing every of the rim edge of Wn.

Definition 2.2.3 (Deb and Limaye[35]). Shell Sn is the graph constructed as a cycle

Cn with (n − 3) chords sharing a common end point (apex). The shell Sn is same

as fan Fn−1. That is, Sn = Fn−1 = Pn−1 +K1.

Definition 2.2.4 (Ayel and Favaron[20]). Helm Hn is the graph constructed from

the wheel Wn by attaching a pendant edge at every vertex of the rim of Wn.

9


Definition 2.2.5 (Gross and Yellen[17]). Closed helm CHn is the graph constructed

from the helm Hn by joing every pendant vertex to form outer cycle.

Definition 2.2.6 (Gross and Yellen[17]). The web graph Wbn is the graph con-

structed by joining the pendant vertices of helm Hn to form a cycle and then adding

a pendant edge to every vertex of outer cycle.

Definition 2.2.7 (Andar et al.[31]). Flower Fln is the graph constructed from the

helm Hn by attaching every pendant vertex to the apex vertex of helm Hn.

Definition 2.2.8 (Deb and Limaye[35]). The double fan DFn is obtained by Pn +

2K1.

Definition 2.2.9 (Gallian[18]). An edge joining two non-adjacent vertices of cycle

Cn is called chord of cycle Cn.

Definition 2.2.10 (Gallian[18]). Two edges forming a triangle with an edge of the

cycle Cn(n ≥ 5) are called twin chords of a cycle Cn, it is deoted as Cn,3.

Definition 2.2.11 (Gallian[18]). Cycle Cn with three chords which by themselves

form a triangle are called cycle with triangle, it is denoted as Cn(p, q, r) whose edges

form the edges of cycles Cp+2, Cq+2, Cr+2 without chords, where p, q, r, n ∈ N, n ≥ 6

with p+ q + r + 3 = n.

Definition 2.2.12 (Harary[11]). The graph K1,n is called a star graph in which

d(v0) = n called the apex and d(vi) = 1(1 ≤ i ≤ n).

Definition 2.2.13 (Gallian[18]). Bistar Bm,n is the graph constructed by enlinking

the apex vertices of star K1,m and K1,n by an edge.

Definition 2.2.14 (Gallian[18]). The triangular snake Tn is constructed from the

path Pn by replacing every edge of Pn by triangle C3.

Definition 2.2.15 (Gallian[18]). The double triangular snake DTn includes of two

triangular snakes which have a common path.

Definition 2.2.16 (Gallian[18]). The quadrilateral snake Qn is obtained from the

path Pn by replacing every edge of Pn by cycle C4.

10


Definition 2.2.17 (Gallian[18]). The double quadrilateral snake DQn includes of

two quadrilateral snakes that have a common path.

Definition 2.2.18 (Gallian[18]). An alternate triangular snake A(Tn) is obtained

from a path on vertices v1, v2, . . . , vn by joining vi and vi+1 (alternatively) to a new

vertex ui. i.e Every alternate edge of path is replaced by C3.

Definition 2.2.19 (Gallian[18]). An alternate quadrilateral snake A(Qn) is obtained

from a path on vertices v1, v2, . . . , vn by joining vi, vi+1 (alternatively) to new vertices

ui, wi respectively and then joining ui and wi. i.e Every alternate edge of path is

replaced by C4.

Definition 2.2.20 (Gallian[18]). A double alternate triangular snake DA(Tn) con-

sists of two alternate triangular snakes that have a common path. That is, double

alternate triangular snake is obtained from a path on vertices v1, v2, . . . , vn by joining

vi and vi+1 (alternatively) to new vertices ui and wi.

Definition 2.2.21 (Gallian[18]). A double alternate quadrilateral snake DA(Qn)

consists of two alternate quadrilateral snakes that have a common path. That is, it

is obtained from a path v1, v2, . . . , vn by joining vi and vi+1 (alternatively) to new

vertices ui, u′i and wi, w

′i respectively and adding the edges uiwi and u′iw

′i.

Definition 2.2.22 (Gallian[18]). Comb graph Pn�K1 is the graph constructed from

connecting a pendant edge to every vertex of path Pn.

Definition 2.2.23 (Gallian[18]). Crown graph Cn � K1 is the graph constructed

from connecting a pendant edge to every vertex of cycle Cn.

Definition 2.2.24 (Gross et al.[19]). Armed crown is the graph constructed from

connecting path P2 at every vertex of cycle Cn. It is denoted by ACn, Thus ACn =

Cn � P2.

2.2.2 Operations on a graph / Operations of graphs

Definition 2.2.25. Ringsum of two graphs G1 = (V1, E1) and G2 = (V2, E2), de-

noted as G1 ⊕G2, is the graph G1 ⊕G2 = (V1⋃V2, (E1

⋃E2)− (E1 ∩ E2)).

11


Definition 2.2.26 ([18]). Let G = (V,E) be a graph. Let e = uv be an edge of G

and w be a vertex not in G. The edge e is said to be subdivided when it is replaced

by the edges e′ = uw and e′′ = wv.

Definition 2.2.27. Join of two graphs G1 = (V1, E1) and G2 = (V2, E2), denoted as

G1 +G2, is the graph G1 +G2 = (V1⋃V2, (E1

⋃E2)

⋃{uv | u ∈ V (G1), v ∈ V (G2)}.

Definition 2.2.28. The switching of a vertex v in a graph G means removing all

the edges incident to v and adding edges joining v to every other vertex which is not

adjacent to v in G. The graph constructed from switching of a vertex v in a graph

G is denoted as Gv.

Definition 2.2.29. Two adjacent vertices are called neighbours. The set of all

neighbours of vertex v is called the neighbourhood set of v. It is denoted as N(v) or

N [v] and they are respectively known as open and closed neighbourhood sets.

N(v) = {u ∈ V (G) | u adjacent to v and u 6= v}N [v] = N(v)

⋃{v}

Definition 2.2.30 (Harary[11]). Duplication of a vertex v by a new vertex v′ in a

graph G produces a new graph G′, where v ∈ V (G) and v′ is newly added vertex with

N(v) = N(v′).

Definition 2.2.31 (Harary[11]). Duplication of an edge e = uv by a new edge

e′ = u′v′, in a graph G produces a new graph G′, where u, v ∈ V (G) and e′ = u′v′ is

newly added edge with N(u′) = N(u)⋃{v′} \ {v} and N(v′) = N(v)

⋃{u′} \ {u}.

Definition 2.2.32 (Harary[11]). Duplication of a vertex v by a new edge e′ = v′u′

in a graph G produces a new graph G′, where u, v ∈ V (G) and e′ = u′v′ is newly

added edge with N(v′) = {vk, u′} and N(u′) = {vk, v′}.

Definition 2.2.33 (Harary[11]). Duplication of an edge e = uv by a new vertex v′

in a graph G produces a new graph G′, where u, v ∈ V (G) and v′ is newly added

vertex with N(v′) = {u, v}.

Definition 2.2.34 (Frucht and Harary[46]). Corona G�H of two graphs G and H

is defined as the graph acquired by taking one copy of G (having p1 vertices) and p1

copies of H and joining one copy of H at every vertex of G by an edge.

12

2.3. Graph labeling

2.3 Graph labeling

Alexander Rosa introduced graph labeling. In 1966 he introduced certain valuation

of vertices of the graph which is the origin of most of the graph labeling techniques.

The present work is concerned with graph labeling techniques. Different contempla-

tions of combinatorics and number theory give various graph labeling techniques.

The advancement of research in graph labeling techniques is due to basic labelings

known as graceful and harmonious labeling.

Rosa[6] called an injective function f from V (G) to {0, 1, 2, . . . , q} as β-valuation of

graph G, if every edge xy ∈ G allocated by taking the absolute difference of labels

of end vertices are distinct. Later, Golomb[57] called it graceful labeling. Here q

denotes number of edges in the graph G. Graceful labeling in a graceful graph may

not be unique.

In 1980, Graham and Sloane[42] called an injective function f from V (G) to ({0, 1, 2,. . . , q},+q) as harmonious labeling of a graph G if every edge xy ∈ G allocated by

(f(x) + f(y))(mod q) such that every edge is allocated distinct label.

As a bipoduct of efforts to solve Ringel-Kotzig conjecture which states that “All

trees are graceful”, Cahit[14] illuminated the idea of cordial labeling as a weaker

version of graceful and harmonious labeling. A binary vertex labeling f from V (G)

to {0, 1} of a graph G is known as cordial labeling if every edge xy ∈ G allocated

by taking the absolute difference of labels of end vertices and which satisfies the

conditions |vf (1) − vf (0)| ≤ 1 and |ef (1) − ef (0)| ≤ 1. Here vf (i) and ef (i) are

respectively the number of vertices and number of edges of graph G having label i,

i ∈ {0, 1}.

Different variants of graceful and harmonious labeling were studied by different

authors. There are some open problems and conjectures which are the furthermost

13

2.3. Graph labeling

attraction of this field. Cahit[14] derived some basic cordial graphs such as graphs

Km,n, complete graph Kn iff n ≤ 3, fan graph Fn, wheel graph Wn iff n 6≡ 3( mod 4)

and maximal outer planar graphs. Some labelings such as prime cordial labeling,

product cordial labeling, total product cordial labeling, DC labeling, difference cor-

dial labeling etc. were also introduced as different variants of cordial labeling.

A positive integer p(> 1) is said to be prime if it cannot be written as a prod-

uct of two natural numbers a and b for 1 < a, b < p. Oodles number of qualitative

results, properties and conjectures are available for prime numbers. Two integers a

and b are said to be co-prime if (a, b) = 1.

Sundaram, Ponraj and Somasundaram[32] called a vertex labeling f from V (G)

to {1, 2, . . . , |V (G)|} of a graph G as prime cordial labeling if every edge xy ∈ G,

the function f ∗ from E(G) to {0, 1} is defined as f ∗(xy) = 1 if gcd(f(x), f(y)) = 1

and 0 if gcd(f(x), f(y)) > 1 satisfies the condition |ef (1)) − ef (0)| ≤ 1. The same

author proved that Cn iff n ≥ 6, Pn iff n 6= 3, K1,n (n odd), bistars Bn,n, crowns

Cn �K1 are prime cordial graphs[32].

In 2004, Sundaram, Ponraj and Somasundaram[33] called a binary vertex label-

ing f from V (G) to {0, 1} of a graph G as product cordial labeling if every edge

xy ∈ G, the function f ∗ from E(G) to {0, 1} is defined as f ∗(xy) = f(x)f(y) which

fulfils the conditions |vf (1) − vf (0)| ≤ 1 and |ef (1) − ef (0)| ≤ 1. Here vf (i) is the

number of vertices of graph G having label i and ef (i) is the number of edges of

graph G having label i, i ∈ {0, 1}. In [33], they proved that trees, Cn iff n is odd,

triangular snakes Tn and helms Hn are product cordial graphs.

In 2006, Sundaram[34] called a function f from V (G) to {0, 1} of a graph G as

total product cordial labeling if every edge xy ∈ G, the function f ∗ from E(G) to

{0, 1} is defined as f ∗(xy) = f(x)f(y) and it fulfils the conditions |(vf (1) + ef (1))−(vf (0) + ef (0))| ≤ 1.

14

2.3. Graph labeling

Varatharajan, Navaneethakrishnan and Nagarajan[44] called a function f from V (G)

to {1, 2, . . . , |V (G)|} as DC labeling of a graph G if every edge xy ∈ G identified

the label 1 whenever f(x)|f(y) or f(y)|f(x) and 0 otherwise, which satisfies the

condition |ef (1)−ef (0)| ≤ 1. In the same paper, the authors have derived that path

Pn, cycle Cn, wheel Wn, star K1,n, complete graph Kn for n < 7 and graphs K2,n,

K3,n are DC graphs.

In DC labeling, the edge label for an edge ab is produced by using the condition

“whether a divides b or not”; while in square DC labeling, the edge label for an edge

ab is produced by using the condition “whether a2 divides b or not”. Thus the idea

of square DC labeling differs from DC labeling merely by considering square of one

of the end vertices of the centain edge to produce required edge label.

In 2013, S. Murugesan, D. Jayaraman and J. Shiama[56] called a function f :

V (G)→ {1, 2, . . . , |V (G)|} as square DC labeling of a graph G if every edge xy ∈ Gidentified the label 1 whenever [f(x)]2||f(y) or [f(y)]2||f(x) and 0 otherwise, in such

a way that |ef (1) − ef (0)| ≤ 1. In the same paper, the authors have derived that

path Pn, cycle Cn, wheel Wn, star K1,n and some classes of graph Km,n are square

DC graphs.

In 2015, Kanani and Bosmia[25] called a function f from V (G) to {1, 2, . . . , |V (G)|}as cube DC labeling of a graph G if every edge xy ∈ G is identified the label 1

whenever [f(x)]3||f(y) or [f(y)]3||f(x) and 0 otherwise, it satisfies the condition

|ef (1) − ef (0)| ≤ 1. In the same paper, they have derived that path Pn, cycle Cn,

wheel Wn, star K1,n and some classes of Km,n are square DC graphs. The same

authors have constructed cube DC labeling for complete graph Kn, star graph K1,n,

graphs K2,n and K3,n, bistar Bn,n and restricted square graph of Bn,n.

In 2015, Combining by the thought of DC labeling and odd labeling, Muthaiyan

and Pugalenthi[3] called a function f from V (G) to {1, 3, . . . , 2|V (G)|−1} as vertex

odd DC labeling of a graph G if every edge xy ∈ G is identified the label 1 whenever

15

2.4. Concluding Remarks

f(x)|f(y) or f(y)|f(x) and 0 otherwise, it satisfies the condition |ef (1)− ef (0)| ≤ 1.

In the same paper, results on vertex odd DC labeling for shell Sn, helm Hn, flower

Fln, K2,n, < K(1)1,n, K

(2)1,n > were derived.

In 2016, A. Lourdusamy and F. Patrick[1] called a function f from V (G) to {1, 2, . . . ,|V (G)|} as sum DC labeling of a graph G if every edge xy ∈ G identified the label

1 whenever 2|[f(x) + f(y)] and 0 otherwise, in such a way that |ef (1)− ef (0)| ≤ 1.

In this paper, the same authors have established some basic sum DC graphs such

as path Pn, comb Pn�K1, Star K1,n, graph K2,n, bistar Bn,n, jewel, crown Cn�P1,

flower Fln and gear Gn. The authors have also derived sum DC labeling of the

graphs constructed as a result for some graph operations. Few such sum DC graphs

are K2 +mK1, subdivision of the star, K1,3 ∗K1,n and B2n,n.

2.4 Concluding Remarks

This chapter furnishes elementary definitions, terminology and notations necessary

for the advancement of the content. For any undefined notation and terminology

we refer to Harrary[11], Clark and Holton[21], Gross and Yellen[17], West[9], Bondy

and Murty[16], while for the terms related to number theory, we refer to Burton[10].

The next chapter is aimed to the discussion on results related to DC labeling of

graphs resulted due to some graph operations.

16

CHAPTER 3

Divisor Cordial Labeling With

The Use of Some Graph

Operations

Number Theory is a fascinating subject in mathematics. It has so many interest-

ing concepts. The concepts of primality and divisibility play an important role in

number theory. Divisor cordial labeling is one such labeling of graphs which uses

the concept of divisors of a number. Labeling of a graph is allocation of numbers to

vertices or edges or both graph elements. In divisor cordial labeling, the edge labels

are positive integers which are allocated through the condition of divisibility of the

labels (which are of course positive integers) of end vertices.

3.1 Introduction

For a, b ∈ N, a divides b means there exists k ∈ N such that b = ka. It is denoted as

a | b. If a does not divide b, then it is denoted as a - b. By combining the divisibility

concept in number theory and cordial labeling concept in graph labeling, Varathara-

jan, Navaneethakrishnan and Nagarajan originated the notion called divisor cordial

labeling in 2011.

17

3.2. Some Known Results on DC Labeling

Definition 3.1.1 (Varatharajan et al.[44]). Let G = (V,E) be a simple graph with

order p and size q. Consider a bijection f : V (G) → {1, 2, . . . , |V (G)|} and let the

induced function f ∗ : E(G)→ {0, 1} be defined as

f ∗(uv) =

1; if f(u) | f(v) or f(v) | f(u)

0; otherwise

Then the function f is called a divisor cordial labeling if |ef (0)− ef (1)| ≤ 1.

The divisor cordial labeling is also called DC labeling. A graph which confesses DC

labeling is called DC graph.

Example 3.1.1. DC labeling in wheel graph W5 is demonstrated in the following

Figure 3.1.

1

2

6

53

4

v0

v1

v2

v3 v4

v5

Figure 3.1: DC labeling in W5

Naturally, a graph may have more than one divisor cordial labelings. However if

one such labeling exists, the corresponding graph is divisor cordial.

3.2 Some Known Results on DC Labeling

Divisor cordial labeling was introduced in 2011. Since then, many researchers have

explored this labeling by finding captivating results during last eight years.

In first paper on divisor cordial labeling[44], Varatharajan et al. have established

DC labeling for some basic graphs and derived the following results.

Theorem 3.2.1. The path Pn, the cycle Cn and the wheel Wn are DC.

18

3.2. Some Known Results on DC Labeling

Theorem 3.2.2. Kn is DC for n = 1, 2, 3, 5 and 6.

Theorem 3.2.3. Kn is not DC for n = 4 and n ≥ 7.

Theorem 3.2.4. The complete bipartite graphs K1,n, K2,n and K3,n are DC.

Theorem 3.2.5. The barycentric subdivision of the star K1,n is DC.

Theorem 3.2.6. The bistar Bm,n(m ≤ n) is DC.

The same authors have derived more results in [45] which characterize DC graph.

They have also developed DC labeling in some special classes of trees. These results

are listed below.

Theorem 3.2.7. Every full binary tree is DC.

Theorem 3.2.8. The graph G =< K(1)1,n, K

(2)1,n > is DC graph.

Theorem 3.2.9. The graph G =< K(1)1,n, K

(2)1,n, K

(3)1,n > is DC graph.

Many other researchers have worked on this labeling and derived some benchmark

results. Few such results are stated below.

Theorem 3.2.10 (Vaidya and Shah [49]). Splitting graph S ′(K1,n), S ′(Bn,n) of star

K1,n and bistar Bn,n are DC.

Theorem 3.2.11 (Vaidya and Shah [49]). Degree splitting graph DS(Bn,n) and

shadow graph D2(Bn,n) of bistar Bn,n are DC.

Theorem 3.2.12 (Vaidya and Shah [49]). Restricted square graph B2n,n of bistar

Bn,n is DC.

Theorem 3.2.13 (Vaidya and Shah [49]). The helm graph Hn, the flower graph

Fln and the gear graph Gn are DC.

Theorem 3.2.14 (Vaidya and Shah [49]). The graph constructed from switching

invariance in cycle Cn is DC.

Theorem 3.2.15 (Vaidya and Shah [49]). The graph (Wn)v is DC, where v is rim

vertex.

Theorem 3.2.16 (Vaidya and Shah [49]). The graph (Hn)v is DC where v is apex

vertex.

19

3.3. Some New DC Graphs With the Use of Ringsum Operation

Kanani and Bosmia[26] have constructed DC graphs by considering corona product

of K1 with different graph families. In [27], the same authors have discussed DC

labeling for some results constructed from applying different (graph) operations on

bistar Bm,n.

3.3 Some New DC Graphs With the Use of Ringsum Op-

eration

Ghodasara and Rokad[5] illuminated and derived some fascinating results on cordial

labeling of the graphs by considering ringsum of K1,n with different graph families.

Under the inspiration of this credibility, in the current chapter we demonstrate some

new graphs constructed from the graph operation ringsum for DC labeling.

Remark 3.3.1. Throughout this chapter we consider the ringsum of a graph G with

K1,n by considering any one vertex of G and the apex vertex of K1,n as a common

vertex.

Theorem 3.3.1. Cn ⊕K1,n is DC graph.

Proof. Let V (Cn ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (Cn) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.

Let E(Cn ⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤ n}.

It is to be noted that, |V (Cn ⊕K1,n)| = |E(Cn ⊕K1,n)| = 2n.

Consider a bijection f : V (Cn ⊕K1,n)→ {1, 2, 3, . . . , 2n} defined as below.

f(u1) = 2.

f(v1) = 1.

f(uj) = 2j − 1; 2 ≤ j ≤ n.

f(vk) = 2k; 2 ≤ k ≤ n.

As per this pattern, allocate the vertices such that, for any edge ujuj+1 ∈ E(Cn ⊕K1,n),

f(uj) - f(uj+1), 1 ≤ j ≤ n− 1.

20


Also

f(u1) | f(vk)

for each k, 1 ≤ k ≤ n.

By looking into the above prescribed pattern,

ef (1) = ef (0) = n.

Then we get, |ef (1)− ef (0)| ≤ 1 .

That is, Cn ⊕K1,n is DC graph.

Example 3.3.1. DC labeling in C5 ⊕K1,5 is demonstrated in the following Figure

3.2.

v3 v5v1 v2 v4

1

3

2

4

5

6

7

8

9

10

u1

u2

u3

u5

u4

Figure 3.2: DC labeling in C5 ⊕K1,5

Theorem 3.3.2. G⊕K1,n is DC graph, where G is the cycle Cn with one chord.

Proof. Let G denote the cycle Cn with one chord.

Let V (G ⊕ K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (G) = {uj | 1 ≤ j ≤ n} and

V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.

Let E(G⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n−1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤ n}⋃{u2un},

where u2un is the chord of Cn and vertices u1, u2, un form a triangle with chord u2un.

It is to be noted that, |V (G⊕K1,n)| = 2n and |E(G⊕K1,n)| = 2n+ 1.

21


Consider a bijection f : V (G⊕K1,n)→ {1, 2, 3, . . . , 2n} defined as below.

f(u1) = 2.

f(v1) = 1.

f(uj) = 2j − 1; 2 ≤ j ≤ n.

f(vk) = 2k; 2 ≤ k ≤ n.

As per this pattern, label the vertices such that, for any edge ujuj+1 ∈ E(G⊕K1,n),

f(uj) - f(uj+1), 1 ≤ j ≤ n− 1.

Also

f(u1) | f(vk)

for each k, 1 ≤ k ≤ n.


ef (1) = n, ef (0) = n+ 1.

Hence, |ef (0)− ef (1)| ≤ 1.

That is, G⊕K1,n is DC graph, where G is the cycle Cn with one chord.

Example 3.3.2. DC labeling of ringsum in C6 with one chord and K1,6 is demon-

strated in the following Figure 3.3.

v3 v4 v5 v6v1 v2

u3

u4

u5

u6

u1

u2 3

2

5

7

9

11

1 4 6 8 10 12

Figure 3.3: DC labeling in the graph constructed from ringsum of C6 with one chord and K1,6

Theorem 3.3.3. Cn,3⊕K1,n is DC graph, where Cn,3 is the cycle with twin chords.

22


Proof. Let V (Cn,3 ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (Cn,3) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.

Let E(Cn,3 ⊕ K1,n) = {ujuj+1 | 1 ≤ j ≤ n − 1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤

n}⋃{u2un, u2un−1}, where u2un and u2un−1 are the chords of Cn.

It is to be noted that, |V (Cn,3 ⊕K1,n)| = 2n and |E(Cn,3 ⊕K1,n)| = 2n+ 2.

Consider a bijection f : V (Cn,3 ⊕K1,n)→ {1, 2, 3, . . . , 2n} defined as below.

f(uj) = 2j; 1 ≤ j ≤ 2.

f(uj) = 2j + 1; 3 ≤ j ≤ n− 1.

f(un) = 8.

f(vk) = 2k − 1; 1 ≤ k ≤ 3.

f(v4) = 6.

f(vk) = 2k; 5 ≤ k ≤ n.

As per this pattern, label the vertices such that, for any edge ujuj+1 ∈ E(Cn,3⊕K1,n),

f(uj) - f(uj+1), 2 ≤ j ≤ n− 1.

Also

f(u1) | f(vk), j = 1, 4 ≤ k ≤ n

and

f(u1) | f(u2), f(u1) | f(un).


ef (0) = n+ 1 = ef (1).

Then we get, |ef (1)− ef (0)| ≤ 1 .

That is, Cn,3 ⊕K1,n is DC graph, where Cn,3 is the cycle Cn with twin chords.

Example 3.3.3. DC labeling in graph C7,3 ⊕K1,7 is demonstrated in the following

Figure 3.4.

23


u3

u4 u5

u6

u7

u1

u2

v3 v4 v5 v6 v7v1 v2

2

7

9 11

3

4

6

8

10 12 14

13

51

Figure 3.4: DC labeling in C7,3 ⊕K1,7

Theorem 3.3.4. Cn(1, 1, n − 5) ⊕ K1,n is DC graph, where Cn(1, 1, n − 5) is the

cycle with triangle.

Proof. Let G be the cycle with triangle Cn(1, 1, n− 5).



Let E(G⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤ n}⋃{u1u3,

u3un−1, un−1u1}, where u1, u3 and un−1 be the vertices of the triangle formed by the

chords u1u3, u3un−1 and u1un−1.


Consider a bijection f : V (G⊕K1,n)→ {1, 2, 3, . . . , 2n} defined as below.

f(u1) = 2.

f(v1) = 3.

f(u2) = 1.

f(uj) = 2i− 1; 3 ≤ j ≤ n.

f(vk) = 2j; 2 ≤ k ≤ n.

As per this pattern, label the vertices such that, for any edge ujuj+1 ∈ E(G⊕K1,n),

f(uj) - f(uj+1), 3 ≤ j ≤ n− 1.

24


Also

f(u1) | f(vk), 2 ≤ k ≤ n

and

f(u1) | f(u2), f(u2) | f(u3).


ef (0) = n+ 1, ef (1) = n+ 2.

Then we get, |ef (1)− ef (0)| ≤ 1 .

That is, Cn(1, 1, n− 5)⊕K1,n is DC graph.

Example 3.3.4. DC labeling in graph C8(1, 1, 3) ⊕ K1,8 is demonstrated in the

following Figure 3.5.

u3

u4

u5

u6

u7

u1

u2 u8

2

1

5

7

9

11

13

15

3 4 6 8 10 12 1614v3 v4 v5 v6 v7v1 v2 v8

Figure 3.5: DC labeling in C8(1, 1, 3)⊕K1,8

Theorem 3.3.5. Wn ⊕K1,n is DC graph.

Proof. Let V (Wn ⊕K1,n) = {u0, uj, vj | 1 ≤ j ≤ n}, where V (Wn) = {u0, uj | 1 ≤j ≤ n} and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here u0 is the apex vertex, uj(1 ≤ j ≤ n) are rim vertices of Wn and vj(1 ≤ j ≤ n)

are the pendant vertices, u1 is apex vertex of star graph.

Let E(Wn⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n−1}⋃{unu1}⋃{u0uj | 1 ≤ j ≤ n}⋃{u1vj |

1 ≤ j ≤ n}.It is to be noted that, |V (Wn ⊕K1,n)| = 2n+ 1 and |E(Wn ⊕K1,n)| = 3n.

25


Consider a bijection f : V (Wn ⊕K1,n)→ {1, 2, 3, . . . , 2n+ 1} defined as below.

f(u0) = 1.

f(uj) = j + 1; 1 ≤ j ≤ n.

f(vk) = n+ k + 1; 1 ≤ k ≤ n.

As per this pattern, label the vertices such that, for any edge ujuj+1 ∈ E(Wn⊕K1,n),

f(uj) - f(uj+1), 1 ≤ j ≤ n− 2.

Also

f(u0) | f(uj), 1 ≤ j ≤ n.

and

f(u1) | f(vk) whenever k is even, 1 ≤ k ≤ n.


Cases of n Edge conditions

n ≡ 0, 2(mod 4) ef (1) = 3n2

= ef (0)

n ≡ 1, 3(mod 4) ef (1) = d3n2e, ef (0) = b3n

2c

Then we get, |ef (1)− ef (0)| ≤ 1 .

That is, Wn ⊕K1,n is DC graph.

Example 3.3.5. DC labeling in W5 ⊕K1,5 is demonstrated in the following Figure

3.6.

v3 v5v1 v2 v4

u1

u2

u3

u5

u4

2

4

6

8 10

31

5

7 9

u0

11

Figure 3.6: DC labeling in W5 ⊕K1,5

26


Theorem 3.3.6. Fln ⊕K1,n is DC graph.

Proof. Let V (Fln ⊕K1,n) = {u, uj, vj, wj | 1 ≤ j ≤ n}, where V (Fln) = {u, uj, wj |1 ≤ j ≤ n} and V (K1,n) = {w1, vj | 1 ≤ j ≤ n}.Here u is apex vertex, uj(1 ≤ j ≤ n) are internal vertices and wj(1 ≤ j ≤ n) are

external vertices of Fln and d(vj) = 1, where 1 ≤ j ≤ n, w1 is apex vertex of star

graph.

Let E(Fln ⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{uuj | 1 ≤ j ≤ n}⋃{uwj | 1 ≤ j ≤n}⋃{ujwj | 1 ≤ j ≤ n}⋃{w1vj | 1 ≤ j ≤ n}.It is to be noted that, |V (Fln ⊕K1,n)| = 3n+ 1 and |E(Fln ⊕K1,n)| = 5n.

Consider a bijection f : V (Fln ⊕K1,n)→ {1, 2, 3, . . . , 3n+ 1} defined as below.

f(u) = 1.

f(uj) = 2i+ 1; 1 ≤ j ≤ n.

f(wj) = 2j; 1 ≤ j ≤ n.

Allocate the labels {2n + 2, 2n + 3, . . . , . . . 3n + 1} to the vertices v1, v2, . . . , vn in

any order.

As per this pattern, label the vertices such that, for any edge ujuj+1 ∈ E(Fln⊕K1,n),

f(uj) - f(uj+1), 1 ≤ j ≤ n− 1.

Further

f(u) | f(uj), f(u) | f(wj), 1 ≤ j ≤ n

and

f(w1) | f(vk) whenever k is odd, 1 ≤ k ≤ n.

Then we get, |ef (1)− ef (0)| ≤ 1 .

That is, Fln ⊕K1,n is DC graph.

Example 3.3.6. DC labeling in Fl4 ⊕K1,4 is demonstrated in the following Figure

3.7.

27


u1

u2

u3

u4

v1

v2 v3

v4

u1

3

7

954

6

8

2

10

11 12

13

w2

w1

w3

w4

Figure 3.7: DC labeling in Fl4 ⊕K1,4

Theorem 3.3.7. Sn ⊕K1,n is DC graph.

Proof. Let V (Sn ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (Sn) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.

Let E(Sn⊕K1,n) = {ujuj+1, unu1 | 1 ≤ j ≤ n−1}⋃{u1uj | 3 ≤ j ≤ n−1}⋃{u1vj |1 ≤ j ≤ n}.It is to be noted that, |V (Sn ⊕K1,n)| = 2n and |E(Sn ⊕K1,n)| = 3n− 3.

Consider a bijection f : V (Sn ⊕K1,n)→ {1, 2, 3, . . . , 2n} defined as below.

f(u1) = 2.

For the vertices u2, u3, . . . , uk, allocate the vertex labels as per the below ordered

pattern upto it generate k edges with label 1.

1, 2× 21, 2× 22 . . . , 2× 2k1 ,

3, 3× 21, 3× 22 . . . , 3× 2k2 ,

5, 5× 21, 5× 22 . . . , 5× 2k3 ,

. . . , . . . , . . . , . . . ,

. . . , . . . , . . . , . . .

Observe that (2m− 1)2km ≤ n and km ≥ 0 (m ≥ 1).

(2m− 1)2α | (2m− 1)2α+1 and (2m− 1)2ki - 2m+ 1.

28


Then for remaining vertices of uk+1, uk+2, . . . , un, allocate the vertex labels such that

the consecutive vertices do not generate edge label 1.

f(vj) = f(un) + j; 1 ≤ j ≤ n.


Cases of n Edge condition

n ≡ 0, 2(mod 4) ef (0) =⌈3n−1

2

⌉, ef (1) =

⌊3n−1

2

⌋

n ≡ 1, 3(mod 4) ef (1) = 3n−12

= ef (0)

Then we get, |ef (1)− ef (0)| ≤ 1 .

That is, Sn ⊕K1,n is DC graph.

Example 3.3.7. DC labeling in the graph S7⊕K1,7 is demonstrated in the following

Figure 3.8.

u3

u4u5

u6

u7

u1

u2

v3 v4 v5 v6 v7v1 v2

5

9 13

6

10 12 1411

1

2

7

8

3

4

Figure 3.8: DC labeling in S7 ⊕K1,7

Theorem 3.3.8. Pn ⊕K1,n is DC graph.

Proof. Let V (Pn ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (Pn) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.

Let E(Pn ⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{u1vj | 1 ≤ j ≤ n}.It is to be noted that, |V (Pn ⊕K1,n)| = 2n and |E(Pn ⊕K1,n)| = 2n− 1.

29


Consider a bijection f : V (Pn ⊕K1,n)→ {1, 2, 3, . . . , 2n} defined as below.

f(u1) = 2.

f(v1) = 1.

f(uj) = 2j − 1; 2 ≤ j ≤ n.

f(vk) = 2k; 2 ≤ k ≤ n.

As per this pattern, label the vertices such that, for any edge ujuj+1 ∈ E(Pn⊕K1,n),

f(uj) - f(uj+1), 1 ≤ j ≤ n− 1.

Also

f(u1) | f(vk) 1 ≤ k ≤ n

.


ef (1) = n, ef (0) = n− 1.

Then we get, |ef (1)− ef (0)| ≤ 1 .

That is, Pn ⊕K1,n is DC graph.

Example 3.3.8. DC labeling in graph P5 ⊕ K1,5 is demonstrated in the following

Figure 3.9.

u1u2u3u4u5

v1

v2

v3

v4

v5

3 2579

14

6

8

10

Figure 3.9: DC labeling in P5 ⊕K1,5

Theorem 3.3.9. DFn ⊕K1,n is DC graph.

Proof. Let V (DFn ⊕K1,n) = {u,w, uj, vj | 1 ≤ j ≤ n}, where V (DFn) = {u,w, uj |1 ≤ j ≤ n} and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.

30


Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.

Let E(DFn⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{uvj | 1 ≤ j ≤ n}⋃{uuj | 1 ≤ j ≤n}⋃{wuj | 1 ≤ j ≤ n}.It is to be noted that, |V (DFn ⊕K1,n)| = 2n+ 2 and |E(DFn ⊕K1,n)| = 4n− 1.

Consider a bijection f : V (DFn ⊕K1,n)→ {1, 2, 3, . . . , 2n+ 2} defined as below.

f(u) = 1.

f(w) = p, where p is highest prime number ≤ 2n+ 2.

f(v1) = 2.

f(u1) = 3.

f(uj) = 2j; 2 ≤ j ≤ n.

f(vk) = 2k + 1; 2 ≤ k ≤ n− 1.

f(vn) = 2n+ 2.

As per this pattern, label the vertices such that, for any edge ujuj+1 ∈ E(DFn ⊕K1,n),

f(uj) - f(uj+1), 1 ≤ j ≤ n− 1.

Also

f(u) | f(vk), 1 ≤ k ≤ n

and

f(u) | f(uj), f(w) - f(uj) 1 ≤ j ≤ n.


ef (1) = 2n, ef (0) = 2n− 1.

Then we get, |ef (1)− ef (0)| ≤ 1 .

That is, DFn ⊕K1,n is DC graph.

Example 3.3.9. DC labeling in graph DF5 ⊕K1,5 is demonstrated in the following

Figure 3.10.

31


u1 u2 u3 u4 u5

79

v1

v2

v3

v4

v5

u

w

1

3 4 6 8 10

5

12

11

2

Figure 3.10: DC labeling in DF5 ⊕K1,5

Theorem 3.3.10. K2,n ⊕K1,n is DC graph.

Proof. Let V (K2,n ⊕K1,n) = {u,w, uj, vj | 1 ≤ j ≤ n}, where V (K2,n) = {u,w, uj |1 ≤ j ≤ n} and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.

Let E(K2,n ⊕K1,n) = {uuj | 1 ≤ j ≤ n}⋃{wuj | 1 ≤ j ≤ n}⋃{u1vj | 1 ≤ j ≤ n}.It is to be noted that, |V (K2,n ⊕K1,n)| = 2n+ 2 and |E(K2,n ⊕K1,n)| = 3n.

Consider a bijection f : V (K2,n ⊕K1,n)→ {1, 2, 3, . . . , 2n+ 2} defined as below.

f(u) = 1.

f(w) = p,where, p = max {x | x is the largest prime number x ≤ 2n+ 2}.

f(uj) = j + 1; 1 ≤ j ≤ n.

Allocate the labels {n+2, n+3, . . . , p−1, p+1, . . . 2n+2} to the vertices v1, v2, . . . , vn

in any order.

As per this pattern, label the vertices such that, for any edge e ∈ E(K2,n ⊕K1,n),

f(u) | f(uj), f(w) - f(uj) 1 ≤ j ≤ n.

Also

f(u1) | f(vk) whenever k is even, 1 ≤ k ≤ n.

As per the above stated labeling pattern,

32

3.4. DC Labeling With the Use of Switching Invariance in Cycle Allied Graphs


n ≡ 0, 2(mod 4) ef (1) = 3n2

= ef (0)

n ≡ 1, 3(mod 4) ef (1) =⌈3n2

⌉, ef (0) =

⌊3n2

⌋

Then we get, |ef (1)− ef (0)| ≤ 1 .

That is, K2,n ⊕K1,n is DC graph.

Example 3.3.10. DC labeling in K2,7⊕K1,7 is demonstrated in the following Figure

3.11

u1

u2

v1

v2

u3

v3 v4

v5

u4

u

u5

u6

u7

v6

v7

1

2

13

5

7

9

3

4

6

8

16

15

141211

10

w

Figure 3.11: DC labeling in K2,7 ⊕K1,7

Remark 3.3.2. In each of the above theorems, for the ringsum operation with K1,n,

one can consider any arbitrary vertex of the graph G under consideration and by

different permutations of the vertex labels provided in the respective labeling pattern,

one can easily check that the resultant graph still admit DC labeling.

3.4 DC Labeling With the Use of Switching Invariance in

Cycle Allied Graphs

Vaidya and Shah[49] derived some captivating results on DC labeling of the graphs

constructed from switching a vertex in different graphs. In the current section we

demonstrate few graphs constructed from switching invariance in cycle allied graphs

for DC labeling.

33


Theorem 3.4.1. Gv is DC, where G is gear Gn graph and v is not apex vertex.

Proof. Let V (Gn) = {uj | 0 ≤ j ≤ 2n}, where u0 is the apex vertex and uj(1 ≤ j ≤2n) are vertices of gear graph Gn such that

deg(ui) =

2; if j is even.

3; if j is odd.

Let E(Gn) = {u0u2j−1 | 1 ≤ j ≤ n}⋃{ujuj+1 | 1 ≤ j ≤ 2n− 1}⋃{u2nu1}.(Gn)ui

∼= (Gn)uj , where d(ui) = d(uj).

Let (Gn)uj denote the graph constructed from switching of vertex uj (j = 1, 2) of

Gn.

Corresponding to the vertices of different degree in Gn, it is required to discuss

following two cases.

Case 1: d(u1) = 3.

Then by the effect of switching operation, the edge set of (Gn)u1 is

E((Gn)u1) = {u0u2j−1 | 2 ≤ j ≤ n}⋃{ujuj+1 | 2 ≤ j ≤ 2n − 1}⋃{u1uj | 3 ≤ j ≤2n− 1}.Here note that, |V ((Gn)u1)| = 2n+ 1 and |E((Gn)u1)| = 5n− 6.

Consider a bijection f : V ((Gn)u1)→ {1, 2, 3, . . . , 2n+ 1} defined as below.

Our aim is to generate b5n−62c edges with label 1 and d5n−6

2e edges with label 0.

Let f(u0) = p, where p is the largest prime, p ≤ 2n+ 1.

Then n− 1 edges with label 0 will be generated.

Assign f(u1) = 1, which generates 2n− 3 edges with label 1.

Now it remains to generate k = b5n−62c − (2n− 3) edges with label 1.

For the vertices u2, u3, . . . , u2n, assign the vertex labels as per the following ordered


2, 2× 21, 2× 22 . . . , 2× 2k1 ,

3, 3× 21, 3× 22 . . . , 3× 2k2 ,

5, 5× 21, 5× 22 . . . , 5× 2k3 ,

. . . , . . . , . . . , . . . ,

34


. . . , . . . , . . . , . . .


(2m− 1)2α | (2m− 1)2α+1 and (2m− 1)2ki - 2m+ 1.

Then for remaining vertices of (Gn)u1 , assign the vertex labels such that the consec-

utive vertices do not generate edge label 1.



n ≡ 0, 2(mod 4) ef (1) = 5n−62

= ef (0)

n ≡ 1, 3(mod 4) ef (1) = 5n−52, ef (0) = 5n−7

2

Then we get, |ef (1)− ef (0)| ≤ 1 in this case.

Case 2: d(u2) = 2


E((Gn)u2) = {u0u2i−1 | 1 ≤ i ≤ n}⋃{uiui+1 | 3 ≤ i ≤ 2n − 1}⋃{u2nu1}⋃{u2ui |

4 ≤ i ≤ 2n}.Here note that, |V (Gn))u2| = 2n+ 1 and |E(Gn))u2| = 5n− 4.

Using the same labeling pattern as in Case 1, we get the following.


n ≡ 0, 2(mod 4) ef (1) = 5n−42

= ef (0)

n ≡ 1, 3(mod 4) ef (1) = 5n−32, ef (0) = 5n−5

2


That is, (Gn)v is DC, v is not apex vertex.

Example 3.4.1. The following Figure 3.12 demonstrates

(i) Gear graph G6.

(ii) DC labeling in (G6)v, where d(v) = 3.

(iii) DC labeling in (G6)v, where d(v) = 2.

35


u0

u2

u3

u4

u5

u6

u7

u8

u9

u10

u12

u1

u11

13

7

1

5

9

2

3

4

6

8 10

11

122

37

4

6

9

10

8

1112

13

1

5

Figure 3.12: DC labeling in (G6)v

Theorem 3.4.2. Gv is DC, where G is shell graph Sn and v is not apex vertex.

Proof. Let V (Sn) = {uj | 0 ≤ j ≤ n − 1}, where u0 is the apex vertex and uj(1 ≤j ≤ n− 1) are vertices of shell graph Sn, where

deg(uj) =

2; if j = 1 and n− 1.

3; if j = 2, 3, . . . , n− 2.

Let E(Sn) = {u0uj | 2 ≤ j ≤ n− 2}⋃{ujuj+1 | 0 ≤ j ≤ n− 2}⋃{u0un−1}.(Sn)ui

∼= (Sn)uj , where d(ui) = d(uj).

Let (Sn)uj denote the graph constructed from switching of vertex uj (j = 1, 2) of

Sn.

Corresponding to the vertices of different degree in Sn, it is required to discuss fol-

lowing two cases.

Case 1: d(u2) = 3.

Then by the effect of switching operation, the edge set of (Sn)u2 is

E((Sn)u2) = {u0uj | 3 ≤ j ≤ n−2}⋃{ujuj+1 | 3 ≤ j ≤ n−2}⋃{un−1u0}{u0u1}⋃{u2uj |

4 ≤ j ≤ n− 1}.Here note that, |V ((Sn)u2)| = n and |E((Sn)u2)| = 3n− 10.

Consider a bijection f : V ((Sn))u2)→ {1, 2, 3, . . . , n} defined as below.

Subcase 1: n = 6.

f(uj) = j + 1; j = 0, 1, 2.

f(uj) = j; j = 4.

36


f(u3) = p, where p = max {x | x is a prime number and x ≤ n}.

f(un−1) = n.

Subcase 2: n 6= 6, n ∈ N.

Label the vertices u0, u1, . . . un−1 as per the following pattern.

1, 1× 21, 1× 22 . . . , 1× 2k1 ,

3, 3× 21, 3× 22 . . . , 3× 2k2 ,

5, 5× 21, 5× 22 . . . , 5× 2k3 ,

. . . , . . . , . . . , . . . ,

. . . , . . . , . . . , . . .


(2m− 1)2α | (2m− 1)2α+1 and (2m− 1)2ki - 2m+ 1.



n ≡ 0, 2(mod 4) ef (1) = 3n−102

= ef (0)

n ≡ 1, 3(mod 4) ef (1) = 3n−92, ef (0) = 3n−9

2


Case 2: d(u1) = 2.


E((Sn)u1) = {u0uj | 2 ≤ j ≤ n − 1}⋃{ujuj+1 | 2 ≤ j ≤ n − 1}⋃{un−1u0}{u1uj |3 ≤ j ≤ n− 1}.Here note that, |V ((Sn)u1)| = n and |E((Sn)u1)| = 3n− 8.

Let us f : V ((Sn)u1)→ {1, 2, 3, . . . , n} defined as below

f(uj) = j + 1; 0 ≤ j ≤ n− 1.


37



n ≡ 0, 2(mod 4) ef (1) = 3n−82

= ef (0)

n ≡ 1, 3(mod 4) ef (1) = 3n−72, ef (0) = 3n−9

2


That is, (Sn)v is DC, v is not apex vertex.


(i) Shell graph S7.

(ii) DC labeling in (S7)v, where d(v) = 3.

(iii) DC labeling in (S7)v, where d(v) = 2.

2

1

4

3

5

6

7

u2

u3 u4

u5

u1

u0

u6

2

1

3

4 5

6

7

Figure 3.13: DC labeling in (S7)v

Theorem 3.4.3. Gv is DC, where G is flower graph Fln and v is not apex vertex.

Proof. Let V (Fln) = {uj | 0 ≤ j ≤ n}⋃{vj | 1 ≤ j ≤ n}, where d(u0) = 2n,

d(uj) = 4(1 ≤ j ≤ n) are the internal vertices and d(vj) = 2(1 ≤ j ≤ n) are the

external vertices.

Let E(Fln) = {ujuj+1 | 1 ≤ j ≤ n − 1}⋃{u0uj | 1 ≤ j ≤ n}⋃{u0vj | 1 ≤ j ≤n}⋃{ujvj | 1 ≤ j ≤ n}.(Fln)ui

∼= (Fln)uj , where d(ui) = d(uj).

Let (Fln)u1 and (Fln)v1 denote the graph constructed from switching of vertex u1

and v1 of Fln respectively.

Corresponding to the vertices of different degree in Fln, it is required to discuss


38


Case 1: d(u1) = 4.

Then by the effect of switching operation, the edge set of (Fln)u1 is

E((Fln)u1) = {ujuj+1 | 2 ≤ j ≤ n}⋃{u0uj | 2 ≤ j ≤ n}⋃{u0vj | 1 ≤ j ≤n}⋃{ujvj | 2 ≤ j ≤ n}⋃{u1uj | 3 ≤ j ≤ n− 1}⋃{u1vj | 2 ≤ j ≤ n}.Here note that, |V ((Fln)u1)| = 2n+ 1 and |E((Fln)u1)| = 6n− 8.

Consider a bijection f : V ((Fln)u1)→ {1, 2, 3, . . . , 2n+ 1} defined as below.

f(u0) = 1.

f(uj) = 2j; 1 ≤ j ≤ n.

f(vk) = 2k + 1; 1 ≤ j ≤ n.


ef (1) = 3n− 4 = ef (0).

Then we get, |ef (1)− ef (0)| ≤ 1, in this case.

Case 2: d(v1) = 2.

Then by the effect of switching operation, the edge set of (Fln)v1 is

E((Fln)v1) = {ujuj+1 | 1 ≤ j ≤ n}⋃{u0uj | 1 ≤ j ≤ n}⋃{u0vj | 2 ≤ j ≤n}⋃{ujvj | 2 ≤ i ≤ n}⋃{v1uj | 2 ≤ j ≤ n}⋃{v1vj | 2 ≤ j ≤ n}.Also it is to be noted that, |V ((Fln)v1)| = 2n+ 1 and |E((Fln)v1)| = 6n− 4.

Let us a function f from ((Fln)v1) to {1, 2, 3, . . . , 2n+ 1} defined as below.

f(u0) = 1.

f(uj) = 2j + 1; 1 ≤ j ≤ n.

f(vk) = 2k; 1 ≤ k ≤ n.


ef (1) = 3n− 2 = ef (0).

Then we get, |ef (1)− ef (0)| ≤ 1, in this case.

That is, (Fln)v is DC, v is not apex vertex.

39



(i) Flower graph Fl4.

(ii) DC labeling in (Fl4)v1, where d(v1) = 2.

(iii) DC labeling in (Fl4)u1, where d(u1) = 4.

6

2

1

8

9

7

3

4

5

v1v4

u1 u4

v2v3

u2 u3

u0

3

4 6

8

9

75

1

2

Figure 3.14: DC labeling in (Fl4)v

Remark 3.4.1. The graph constructed from switching of a pendant vertex in star

K1,n is isomorphic to K2,n−1 and hence confess DC labeling (Refer [44]).

Remark 3.4.2. (K2,n)v is DC graph because of the following two possibilities.

1. Switching of a vertex with degree 2 in K2,n is isomorphic to K3,n−1 and confess

DC labeling (Refer [44]).

2. Switching of a vertex with degree n in K2,n is isomorphic to K1,n+1 and hence

confess DC labeling (Refer [44]).

Remark 3.4.3. (K3,n)v is DC graph because of the following two possibilities.

1. Switching of a vertex with degree three in K3,n is isomorphic to K4,n−1 and

hence confess DC labeling (Refer [44]).

2. Switching of a vertex with degree n in K3,n is isomorphic to K2,n+1 and hence

confess DC labeling (Refer [44]).

40

3.5. DC Labeling With the Use of Duplication of a Vertex/Edge in Star Graph

3.5 DC Labeling With the Use of Duplication of a Ver-

tex/Edge in Star Graph

Vaidya and Prajapati[55] derived some attractive results on prime labeling of graphs

constructed from duplication of graph elements. In this section we demonstrate some

DC graphs constructed from duplication of vertex/edge in star K1,n.

Theorem 3.5.1. The graph constructed from duplication of a vertex in star K1,n is

DC.

Proof. Let V (K1,n) = {v0, vj | 1 ≤ j ≤ n}, where v0 is the apex vertex and

d(vj) = 1(1 ≤ j ≤ n). E(K1,n) = {v0vj | 1 ≤ j ≤ n}.Let G denote the resultant graph constructed from duplication of any vertex vj by

vertex v′j in K1,n.

Corresponding to the vertices of different degree in K1,n, it is required to discuss


Case 1: d(v0) = n.

The graph constructed from duplication of apex vertex v0 in K1,n is the graph K2,n

and hence it is DC graph (Refer [44]).

Case 2: d(v1) = 1.

The graph constructed from duplication of any pendant vertex in K1,n is a star

graph K1,n+1 and hence it is DC graph (Refer [44]).

Thus in each case the proof is an immidiate outcome of the result proved by

Varatharajan, Navanaeethakrishnan and Nagarajan [44].

Theorem 3.5.2. The graph constructed from duplication of a vertex by an edge in

star K1,n is DC.

Proof. Let V (K1,n) = {vj | 0 ≤ j ≤ n}, where v0 is the apex vertex and vj are the

pendant vertices, j = 1, 2, . . . , n.

Let E(K1,n) = {v0vj | 1 ≤ j ≤ n}.Let G denote the resultant graph constructed from duplicating an arbitrary vertex

vj by an edge v′jv′′j in K1,n.

41


It is to be noted that, |V (G)| = n+ 3 = |E(G)|.Corresponding to the vertices of different degree in K1,n, it is required to discuss

below two possibilities.

Case 1: d(v0) = n.

Let us consider the vertex be v0 and it’s duplicated edge be v′0v′′0 .

.Let us a function f from V (G) to {1, 2, 3, . . . , n+ 3} defined as below.

f(v0) = 2.

f(v1) = 1.

f(v′0) = n+ 2.

f(v′′0) = n+ 3.

f(vj) = j + 1; 2 ≤ j ≤ n.


Case 2: d(vj) = 1.

Let us consider the vertex be vj and it’s duplicated edge be v′jv′′j .

WLOG we may assume that vj = vn.

Let us a function f from V (G) to {1, 2, 3, . . . , n+ 3} defined as below.

f(v0) = 2.

f(v′n) = n+ 3.

f(v′′n) = 1.

f(vj) = j + 2; 1 ≤ j ≤ n.

Then we get, |ef (1)− ef (0)| ≤ 1 .

That is, the graph constructed from duplicating an arbitrary vertex by an edge in

K1,n is DC.

Example 3.5.1. DC labeling of the graphs constructed from duplication of apex ver-

tex v0 by an edge v′0v′′0 and pendant vertex v5 by an edge v′5v

′′5 in K1,5 are demonstrated

in the following Figure 3.15.

42


v5

1

2

3

5

6 7

8

4

v0

v1

v2

v3 v4

v5v0

v1

v2

v3 v4

v5'

1

87v5

v5"

65

3

4

2

v0

v0'

v0"

v1

v2

v3

v4

Figure 3.15: DC labeling in the graph constructed from duplication of vertex by edge in K1,5

Corollary 3.5.1. The graph constructed from duplication of an edge by a vertex in

star K1,n is DC.

Proof. The graph constructed from duplicating an edge by a vertex in star K1,n is

same as the graph constructed from duplicating apex vertex by an edge; which is a

DC graph (Refer Theorem 3.5.2).

Theorem 3.5.3. The graph constructed from duplication of an edge in star K1,n is

DC.

Proof. Let V (K1,n) = {vj | 0 ≤ j ≤ n}, where v0 is the apex vertex and vj are the

pendant vertices, j = 1, 2, . . . , n.

Let E(K1,n) = {v0vj | 1 ≤ j ≤ n}.Let G denote the resultant graph constructed from duplicating an edge v0vn by a

new edge v′0v′n in K1,n with edge set E(G) = {v0vj | 1 ≤ j ≤ n}⋃{v′0vj | 1 ≤ j ≤

n}⋃{v′0v′n}.Here note that, |V (G)| = n+ 3 and |E(G)| = 2n.

Consider a bijection f : V (G)→ {1, 2, . . . , n+ 3} defined as below.

f(v0) = 1.

f(v′0) = p,where p = max {x | x is a prime number and x ≤ n+ 4}.

Allocate the labels {2, 3, . . . , p− 1, p+ 1, . . . , n+ 3} to the vertices v1, v2, . . . , vn, v′n

in any order.

Then we get, |ef (0)− ef (1)| ≤ 1.

43

3.6. Conclusion and Scope for Further Research

That is, the graph constructed from duplicating any edge in K1,n is DC.

Example 3.5.2. DC labeling of the graph constructed from duplication of an edge

v0v8 by edge v′0v′8 in K1,8 is demonstrated in the following Figure 3.16.

1 9v0

v1

v2

v3

v4 v5

v6

v8

v7

v0 v8

v7

v1

v2

v3

v4

v5

v6

v'0v'8

2

1110

7

8

5

6

3

4

Figure 3.16: DC labeling in the graph constructed from duplication of edge v0v8 in K1,8

3.6 Conclusion and Scope for Further Research

In this chapter we have emanated DC labeling for larger graphs constructed from the

standard graphs by means of various graph operations such as ringsum of different

graphs with star graph, switching of vertex in cycle allied graphs and duplication of

vertex in star allied graphs. At the end, we pose some open problems listed below.

Problem 3.6.1. Derive essential and adequate condition (if any) for any graph to

be DC graph.

Problem 3.6.2. To investigate some new DC graphs with respect to other graph

operations.

Problem 3.6.3. Classify/Generalize the graphs G such that G⊕K1,n is DC graph.

(Here |V (G)| may or may not be equal to n.)

The next chapter is intended to discuss square DC, cube DC and vertex odd DC

labeling of graphs.

44

CHAPTER 4

Square Divisor Cordial, Cube

Divisor Cordial and Vertex Odd

Divisor Cordial Labeling of

Graphs

4.1 Introduction

Divisor cordial labeling has been introduced in the earlier chapter while the current

chapter aims to give a transitory collection of few labelings having DC theme.

4.2 Square DC Labeling

4.2.1 Introduction

Inspired by the idea of DC labeling, Murugesan et al.[56] have established a variant of

DC labeling namely square DC labeling of graphs. In a DC graph with DC labeling

f , the edge label for an edge ab is produced by using the condition “whether f(a)

divides f(b) or not”; while in the square DC labeling, the edge label for an edge ab

is produced by using the condition “whether (f(a))2 divides f(b) or not”. Thus the

concept of square DC labeling differs from DC labeling merely by replacing label of

45

4.2. Square DC Labeling

one of the end vertices of the centain edge by its square to produce required edge

label.

Definition 4.2.1 (Murugesan et al.[56]). Let G = (V,E) be a simple graph with



f ∗(uv) =

1; if [f(u)]2 | f(v) or [f(v)]2 | f(u)

0; otherwise

Then the function f is called a square DC labeling if |ef (0)− ef (1)| ≤ 1.

A graph which confesses square DC labeling is called square DC graph.

Example 4.2.1. A square DC labeling in shell graph S7 is demonstrated in the


v2

v3 v4

v5

v0

v6

1

2

4

3

5

6

7v1

Figure 4.1: Square DC labeling in S7

It is easy to observe that a graph may admit more than one square DC labeling.

However, if one such labeling exists, then the graph becomes square DC graph.

4.2.2 Some Known Results on Square DC Labeling

In first paper on square DC labeling[56], Murugesan et al. have derived square DC

labeling for some basic graphs and derived the following results.

Theorem 4.2.1. The path Pn is square DC iff n ≤ 12.

Theorem 4.2.2. The cycle Cn is square DC iff 3 ≤ n ≤ 11.

46

4.2. Square DC Labeling

Theorem 4.2.3. The wheel graph Wn is square DC.

Theorem 4.2.4. The graph K1,n is square DC iff n = 2, 3, 4, 5 or 7.

Theorem 4.2.5. The graph K2,n is square DC.

Theorem 4.2.6. The graph K3,n is square DC iff n = 1, 2, 3, 5, 6, 7 or 9.

Theorem 4.2.7. The complete graph Kn is square DC iff n = 1, 2, 3 or 5.

Vaidya and Shah[51] have derived some benchmark results on square DC labeling.

Few such results are stated below.

Theorem 4.2.8. The flower graph Fln and bistar Bn,n are square DC.

Theorem 4.2.9. Restricted B2n,n graph and D2(Bn,n) are square DC.

Theorem 4.2.10. Splitting graph of star K1,n and Bistar Bn,n are square DC.

Theorem 4.2.11. Degree splitting graph of bistar Bn,n and path Pn are square DC.

Kanani and Bosmia[28] have discussed square DC labeling for the following graphs

using the graph operation “switching of a vertex”.

Theorem 4.2.12. The graph constructed from switching of a vertex in the bistar

Bm,n and the comb graph Pn �K1 are square DC.

Theorem 4.2.13. (Cn �K1)v and (ACn �K1)v are square DC.

Theorem 4.2.14. The graph constructed from switching of a vertex except apex

vertex in the helm Hn and the gear graph Gn are square DC.

However, there is no such standard relation between either of the two labelings; like,

we may find a graph which admits one labeling but not the other. To observe this

matter more effectively, the list of graph families satisfying/ not satisfying certain

labeling is shown below.

• The wheel graph Wn admits both DC and square DC labeling (Refer [44] and

[56]).

• The path graph P13 is DC graph but not square DC (Refer [44] and [56]).

• The complete graph K7 is neirher DC nor square DC graph (Refer [44] and

[56]).

47

4.3. Cube DC Labeling

4.3 Cube DC Labeling

4.3.1 Introduction

Motivated by two concepts, DC labeling and square DC labeling, Kanani and

Bosmia[25] have established another variant of DC labeling namely cube DC la-

beling. Cube DC labeling can be considered as an extension of square DC labeling

by considering cube of label of one of the end vertex of a certain edge.

Definition 4.3.1 (Kanani and Bosmia[25]). Let G = (V,E) be a simple graph with



f ∗(uv) =

1; if [f(u)]3 | f(v) or [f(v)]3 | f(u)

0; otherwise

Then the function f is called a CDC labeling if |ef (0)− ef (1)| ≤ 1.

A graph which confesses CDC labeling is called CDC graph.

Example 4.3.1. A CDC labeling in graph K2,7 is demonstrated in the following

Figure 4.2.

u1 u2 u3 u4 u5 u6 u7

2 5 73 4 6 8

u1 13

w

Figure 4.2: CDC labeling in K2,7

CDC labeling in a graph (if exists) may not be unique.

48

4.4. Vertex Odd DC Labeling

4.3.2 Some Known Results on Cube DC Labeling

In two different research papers on CDC labeling[25, 26], Kanani and Bosmia have

established CDC labeling for some basic graphs and derived the following attractive

results.

Theorem 4.3.1. The path Pn is CDC graph iff n ≤ 6, n = 8.

Theorem 4.3.2. The cycle Cn is CDC graph iff n = 3, 4, 5.

Theorem 4.3.3. The wheel Wn and flower graph Fln are CDC graphs.

Theorem 4.3.4. The fan graph Fn is CDC graph for all n.

Theorem 4.3.5. Kn is CDC graph iff n ≤ 4.

Theorem 4.3.6. The star graph K1,n is a CDC graph if n ≤ 3.

Theorem 4.3.7. The graph K2,n is a CDC graph.

Theorem 4.3.8. The graph K3,n is CDC if n = 1, 2.

Theorem 4.3.9. The bistar Bn,n and restricted B2n,n are CDC graphs.

However, there is no such standard relation between either of the two square DC

and CDC labelings; like, we may find a graph which admits one labeling but not the

other. To observe this matter more effectively, the list of graph families satisfying/

not satisfying certain labeling is shown below.

• The wheel graph Wn admits both square DC and CDC (Refer [56] and [25]).

• The graph K3,7 is square DC but it is not CDC (Refer [56] and [26]).

• The complete graph K4 is not square DC but it is CDC (Refer [56] and [26]).

• The star graph K1,n(n ≥ 8) is neirher square DC nor CDC (Refer [56] and[26]).

4.4 Vertex Odd DC Labeling

4.4.1 Introduction

Inspired by the idea of DC labeling and odd labeling, Muthaiyan and Pugalenthi[3]

introduced a special type of DC labeling called vertex odd DC labeling.

49


Definition 4.4.1 (Muthaiyan and Pugalenthi[3]). Let G = (V,E) be a simple graph

with order p and size q. Consider a bijection f : V (G) → {1, 3, . . . , 2|V (G)| − 1}and let the edge labeling function f ∗ : E(G)→ {0, 1} be defined as

f ∗(uv) =

1; if f(u) | f(v) or f(v) | f(u)

0; otherwise

Then the function f is called a VODC labeling if |ef (0)− ef (1)| ≤ 1.

A graph which confesses VODC labeling is called VODC graph.

In any VODC graph, VODC labeling may or may not be unique.

Example 4.4.1. A VODC labeling in flower graph Fl7 is demonstrated in the fol-

lowing Figure 4.3.

v0

v1v2

v3

v4

v5

v6

v7

u1

u2

u3

u4

u5

u6

u7

1

3

15

9

5

7

1113

17

19

21

27

29

2325

Figure 4.3: VODC labeling in Fl7

4.4.2 Some Known Results on VODC Labeling

In two different research papers ([3], [4]), Muthaiyan and and Pugalenthi have proved

various graphs to be VODC graphs. These results are stated below.

Theorem 4.4.1. K2,n is VODC.

Theorem 4.4.2. Shell graph Sn is VODC.

50


Theorem 4.4.3. The wheel graph Wn and bistar Bn,n are VODC graphs.

Theorem 4.4.4. The helm Hn and the flower graph Fln are VODC graphs.

Theorem 4.4.5. (Hn)v is a VODC graph, where v is apex vertex.

Theorem 4.4.6. (Pn)v is VODC graph, where v is pendant vertex.

Theorem 4.4.7. (Cn)v is a VODC graph.

Theorem 4.4.8. Barycentric subdivision of K1,n is VODC graph.

Theorem 4.4.9. Restricted square graph, Splitting graph and degree splitting graph

of Bn,n are VODC graph.

Sugumaran and Suresh[48] have discussed further results on VODC labeling of

graphs, few of them are listed below.

Theorem 4.4.10. The gear graph Gn is VODC.

Theorem 4.4.11. The graph P2 +mK1 is a VODC graph.

Theorem 4.4.12. The 1-weak shell graph C(n, n−3) and 2-weak shell graph C(n, n−4) are VODC graphs.

However, there is no such standard relation between either of the two CDC and

VODC labelings; like, we may find a graph which admits one labeling but not the

other. To observe this matter more effectively, the list of graph families satisfying/

not satisfying certain labeling is shown below.

• The wheel graph Wn is both CDC and VODC (Refer [25] and [3]).

• The complete graph K4 is CDC (Refer [26]) but it is not VODC (easy to check).

• The graph W7⊕K1,7 is not CDC (easy to check) but it is VODC (Refer theorem

4.8.5).

• The star graph K1,6 is neirher CDC (Refer [25]) nor VODC (easy to check).

51

4.5. New Results on Square DC, Cube DC and Vertex Odd DC Labeling of Graphs

4.5 New Results on Square DC, Cube DC and Vertex Odd

DC Labeling of Graphs

4.5.1 Introduction

In this section we derive new graphs admitting square DC, CDC and VODC labeling.

Definition 4.5.1 ([18]). A graph G = (V,E) is called a tripartite graph if vertex set

V (G) can be divided into three nonempty disjoint subsets V1, V2 and V3 such that

vertices in the same set are not adjacent to each other.

The complete tripartite graph with |V1| = n1, |V2| = n2, |V3| = n3 is denoted by

Kn1,n2,n3.

Theorem 4.5.1. K1,1,n is square DC graph.

Proof. Let V (K1,1,n) = {u, v, uj | 1 ≤ j ≤ n}, where d(u) = n + 1 = d(v) and

d(uj) = 2(1 ≤ j ≤ n).

Let E(K1,1,n) = {uv, uuj, vuj | 1 ≤ j ≤ n}.It is to be noted that, |V (K1,1,n)| = n+ 2 and |E(K1,1,n)| = 2n+ 1.

Consider a bijection f : V (K1,1,n)→ {1, 2, 3, . . . , n+ 2} defined as below.

f(u) = 1.

f(v) = p,where p = max {x | x is a prime number and x ≤ n+ 2}.

Allocate labels {2, 3, . . . , p − 1, p + 1, . . . , n + 2} to the vertices uj(1 ≤ j ≤ n) of

K1,1,n in any order.

As per this pattern, the vertices are labeled such that

[f(u)]2 | f(uj), 1 ≤ j ≤ n.

Also as f(v) is prime,

[f(v)]2 - f(uj), 1 ≤ j ≤ n.

Further since f(u) = 1,

[f(u)]2 | f(v).

52



ef (1) = ef (0) + 1.

i.e. |ef (0)− ef (1)| ≤ 1.

Therefore the graph under consideration admits square DC labeling and hence K1,1,n

is a square DC graph.

Example 4.5.1. Square DC labeling for the special case of n = 6 in above theorem

is demonstrated in the following Figure 4.4.

1 7u v

u1

u2

u3

u4

u5

u6

6

4

2

3

5

8

Figure 4.4: Square DC labeling in K1,1,6

Corollary 4.5.1. K1,1,n is a CDC graph.

Proof. The labeling function can be defined same as in Theorem 4.5.1. Using the

same arguments considered in above theorem, we have

[f(u)]3 | f(uj), 1 ≤ j ≤ n.

Also as f(v) is prime,

[f(v)]3 - f(uj), 1 ≤ j ≤ n.

Further as f(u) = 1,

[f(u)]3 | f(v).

Hence, K1,1,n is a CDC graph.

Corollary 4.5.2. K1,1,n is a VODC graph.

53


Proof. By considering the similar labeling function as defined in Theorem 4.5.1 and

also using almost same arguments, one can conclude that K1,1,n is a VODC graph.

Corollary 4.5.3. K2 + nK1 is a square DC, CDC and VODC graph.

It is to be noted that: K1,1,n∼= K2 + nK1

Definition 4.5.2 ([18]). Umbrella graph U(m,n) (m > 2 and n > 1) is the graph

constructed from appending a path Pn to the apex of a fan Fm = Pm +K1

Theorem 4.5.2. Umbrella U(n, 3) is a square DC graph.

Proof. Let V (U(n, 3)) = {uj, v1, v2, v3 | 1 ≤ j ≤ n}, where d(v1) = n+ 1, d(v2) = 2,

and d(v3) = 1 and uj(2 ≤ j ≤ n) are the vertices of the path of Fn with v1 as a

central vertex of Fn.

Let E(U(n, 3)) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{v1uj, | 1 ≤ j ≤ n}⋃{v1v2, v2v3}.It is to be noted that, |V (U(n, 3))| = n+ 3 and |E(U(n, 3))| = 2n+ 1.

Consider a bijection f : V (U(n, 3))→ {1, 2, 3, . . . , n+ 3} as per the following.

f(v1) = 1.

f(uj) = j + 1; 1 ≤ j ≤ n.

f(v2) = n+ 2.

f(v3) = n+ 3.


ef (1) = n+ 1, ef (0) = n.

Hence

ef (1) = ef (0) + 1.

Then in each case, we get |ef (0)− ef (1)| ≤ 1.

Therefore umbrella graph U(n, 3) is a square DC graph.

Example 4.5.2. Square DC labeling of the graph U(9, 3)is demonstrated in the


54


u1

u2

u3

u4

u5

u8

u9

u6

u7

v1

v2

v3

1

4

2

3

65 7

8

9

10

11

12

Figure 4.5: Square DC labeling in U(9, 3)

Corollary 4.5.4. Umbrella U(n, 3) is a CDC graph.

Corollary 4.5.5. Umbrella U(n, 3) is a VODC graph.

Definition 4.5.3. [18] A one point union C(t)n of t copies of cycles Cn is the graph

constructed by taking v as a common vertex such that any two cycles C(i)n and

C(j)n (i 6= j) are edge disjoint and do not have any vertex in common except v.

The one point union of t (≥ 1) cycles, each of length n is denoted by C(t)n .

Theorem 4.5.3. C(t)4 is a square DC graph.

Proof. Let V (C(t)4 ) = {vi,j | 1 ≤ i ≤ t, 1 ≤ j ≤ 4}, where v1,1 = v2,1 = v3,1 = v4,1 = v

(say); i.e. v is a common vertex.

Let E(C(t)4 ) = {vi,jvi,j+1 | 1 ≤ i ≤ t, 1 ≤ j ≤ 3}⋃{vi,nvi,1 | 1 ≤ i ≤ t}.

It is to be noted that, |V (C(t)4 )| = 3t+ 1 and |E(C

(t)4 )| = 4t.

Consider a bijection f : V (C(t)4 )→ {1, 2, 3, . . . , 3t+ 1} is defined as below.

f(v) = 1.

f(vi,2) = 3i− 1; 1 ≤ i ≤ t.

f(vi,3) = 3i; 1 ≤ i ≤ t.

f(vi,4) = 3i+ 1; 1 ≤ i ≤ t.


ef (1) = 2t = ef (0).

55


Then we get, |ef (0)− ef (1)| ≤ 1.

That is, C(t)4 is a square DC graph.

Example 4.5.3. Square DC labeling in C(5)4 is demonstrated in the following Figure

4.6.

1

42

3

10

9

8

12

11

13

5

6 7 14 15

16

Figure 4.6: Square DC labeling in C(5)4

Corollary 4.5.6. C(t)4 is a CDC graph.

Corollary 4.5.7. C(t)4 is a VODC graph.

Definition 4.5.4 ([37]). Consider k copies of graph G (wheel, star, fan and friend-

ship) say G(1), G(2), . . . , G(k). Then, the graph < G(1), G(2) . . . , G(t) > is constructed

by joining apex vertex of each G(i) and apex of G(i+1) to a new vertex vi, 1 ≤ i ≤ n−1.

Theorem 4.5.4. The graph < K(1)1,n, K

(2)1,n > is a square DC graph.

Proof. Let G denote the graph < K(1)1,n, K

(2)1,n >.

Let V (G) = {y, v(1)i , v(2)i | 0 ≤ i ≤ n}, where v

(j)i are the pendant vertices of K

(j)1,n,

1 ≤ i ≤ n, j = 1, 2.

Let v(1)0 and v

(2)0 be the apex vertices of K

(1)1,n and K

(2)1,n respectively which are adjacent

to a new common vertex y.

Let E(G) = {v(1)0 v(1)i , v

(2)0 v

(2)i , yv

(1)0 , yv

(2)0 | 1 ≤ i ≤ n}.

It is to be noted that, |V (G)| = 2n+ 3 and |E(G)| = 2n+ 2.

Consider a bijection f : V (G)→ {1, 2, 3, . . . , 2n+ 3} is defined as below.

f(v(1)0 ) = 1.

f(v(2)0 ) = p,where p = max {x | x is a prime number and x ≤ 2n+ 3}.

56


Allocate labels {2, 3, . . . , p − 1, p + 1, . . . , w} to the vertices v(1)1 , v

(1)2 , v

(1)3 , . . . v

(1)n ,

v(2)1 , v

(2)2 , v

(2)3 , . . . v

(2)n , y in any order, where w = 2n + 2 or 2n + 3 whichever is not

prime.


ef (1) = n+ 1 = ef (0).

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, < K(1)1,n, K

(2)1,n > is a square DC graph.

Example 4.5.4. Square DC labeling of the graph < K(1)1,5 , K

(2)1,5 > is demonstrated

in the following Figure 4.7.

2

1

4

3 6

5

7

8

10

9

12 11

13

Figure 4.7: Square DC labeling in < K(1)1,5 ,K

(2)1,5 >

Corollary 4.5.8. The graph < K(1)1,n, K

(2)1,n > is a CDC graph.

Corollary 4.5.9. The graph < K(1)1,n, K

(2)1,n > is a VODC graph.

Definition 4.5.5 ([13]). The graph constructed from given graph G by replacing

every edge ei of G by a graph K2,mifor some mi, 1 ≤ i ≤ q is called arbitrary

supersubdivision of G.

Theorem 4.5.5. Arbitrary supersubdivision of K1,n is a square DC graph.

Proof. V (K1,n) = {vj | 0 ≤ j ≤ n}, where v0 is the apex vertex and d(vj) = 1(1 ≤j ≤ n).

Let E(K1,n) = {v0vj | 1 ≤ j ≤ n}.Let G denote the resultant graph constructed from arbitrary supersubdivision of

K1,n.

Then each edge v0vi of K1,n is exchanged by a graph K2,mifor some natural number

mi.

57


Let uij be the vertices of mi vertex section, 1 ≤ i ≤ n, 1 ≤ j ≤ mi.

Let M =n∑i=1

mi.

It is to be noted that, |V (G)| = n+M + 1 and |E(G)| = 2M .

Consider a bijection f : V (G)→ {1, 2, 3, . . . , n+M + 1} is defined as below.

f(v0) = 1.

Label the vertices vi, 1 ≤ i ≤ n by the last n consecutive prime numbers between 1

to n + M + 1 respectively.

Allocate labels {2, 3, . . . , n + M + 1} to the vertices uij, 1 ≤ i ≤ n, 1 ≤ j ≤ mi in

any order.


ef (1) = ef (0).

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, arbitrary supersubdivision of K1,n is a square DC graph.

Example 4.5.5. Square DC labeling of arbitrary supersubdivision of K1,4 is demon-


1

4 23

6

5

7

8

10

9

12

11

13

14

15

16

17

18

19

Figure 4.8: Square DC labeling in arbitrary supersubdivision of K1,4

58


Remark 4.5.1. Arbitrary supersubdivision of K1,n is nothing but one point union

of the graphs K2,mi, where mi is arbitrary, 1 ≤ i ≤ n.

Corollary 4.5.10. Arbitrary supersubdivision of K1,n is a CDC graph.

Corollary 4.5.11. Arbitrary supersubdivision of K1,n is a VODC graph.

Theorem 4.5.6. The graph constructed from duplication of an edge in K1,n is a

square DC graph.

Proof. Let V (K1,n) = {vj | 0 ≤ j ≤ n}, where d(v0) = n and d(vj) = 1(1 ≤ j ≤ n).

Let E(K1,n) = {v0vj | 1 ≤ j ≤ n}.Let G denote the resultant graph constructed from duplication of the edge e = v0vn

by a new edge e′ = v′0v′n, where

deg(vj) =

n; if j = 0.

1; if j = n.

2; if 1 ≤ j ≤ n− 1.

deg(v′j) =

n; if j = 0.

1; if j = n.

It is to be noted that, |V (G)| = n+ 3 and |E(G)| = 2n.

The labeling f : V (G)→ {1, 2, 3, . . . , n+ 3} is defined as below.

f(v0) = 1.

f(vn) = n+ 2.

f(v′n) = n+ 3.

f(v′0) = p, where p = max{x | x is a prime number and x ≤ n+ 3}.

Allocate labels {2, 3, . . . , p−1, p+1, . . . , n+1} to the vertices v1, v2, . . . , vn−1 in any

order.


ef (1) = n = ef (0).

59


Then we get, |ef (0)− ef (1)| ≤ 1.

That is, the graph constructed from duplication of an edge in K1,n is a square DC

graph.

Example 4.5.6. Square DC labeling of the graph constructed from duplication of

an edge in K1,8 is demonstrated in the following Figure 4.9.

1 9v0

v1

v2

v3

v4 v5

v6

v8

v7

v0 v8

v7

v1

v2

v3

v4

v5

v6

v'0v'8

2

1110

7

8

5

6

3

4

Figure 4.9: Square DC labeling in the graph constructed from duplication of an edge in K1,8

Corollary 4.5.12. The graph constructed from duplication of an edge in K1,n is a

CDC graph.

Corollary 4.5.13. The graph constructed from duplication of an edge in K1,n is a

VODC graph.

Definition 4.5.6 ([45]). Let V (Km,n) = {uj | 1 ≤ j ≤ m}⋃{vj | 1 ≤ j ≤ n}.The

graph Km,n�ui(K1) is defined by connecting a pendant vertex w to the vertex ui for

some i.

Theorem 4.5.7. K2,n � u2(K1) is a square DC graph.

Proof. Let G = K2,n � u2(K1).

Let V (G) = {u1, u2}⋃{vj, w | 1 ≤ j ≤ n}, where w is the pendant vertex adjacent

to u2 in G.

Let E(G) = {u1uj, u2uj, u2w | 1 ≤ j ≤ n}.Also it is to be noted that, |V (G)| = n+ 3 and |E(G)| = 2n+ 1.

60

4.6. VODC Labeling With the Use of Switching of a Vertex in Cycle Allied Graphs

The labeling f from V (G) to {1, 2, 3, . . . , n+ 3} is defined as below.

f(u1) = 1.

f(u2) = p,where p = max {x | x is a prime number and x ≤ n+ 3}.

Allocate labels {2, 3, . . . , p− 1, p+ 1, . . . , w} to the vertices w, vj(1 ≤ j ≤ n) in any

order, where w = n+ 2 or n+ 3 whichever is not prime.


ef (1) = n, ef (0) = n+ 1.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, K2,n � u2(K1) is a square DC graph.

Example 4.5.7. Square DC labeling in the graph K2,5� u2(K1) is demonstrated in

the following Figure 4.10.

1 7

8

42 3 65

u1 u2

v1 v2 v3 v4 v5

w

Figure 4.10: Square DC labeling in K2,5 � u2(K1)

Corollary 4.5.14. K2,n � u2(K1) is a CDC graph.

Corollary 4.5.15. K2,n � u2(K1) is a VODC graph.

4.6 VODC Labeling With the Use of Switching of a Vertex

in Cycle Allied Graphs

In this section, and in next two consecutive sections, VODC labeling of the graphs

constructed from the graph operation “switching of a vertex” is discussed. Particu-

61


larly in this section, we consider the switching of a vertex in cycle allied graphs.

Theorem 4.6.1. The graph Gv, where G is cycle Cn with one chord, is VODC.

Proof. Let G denote the graph cycle Cn with one chord.

Let V (G) = {vj | 1 ≤ j ≤ n} = V (Cn).

Let E(G) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{v2vn}, where v2vn is the chord of

Cn. WLOG let the switched vertex be v1 (of degree 2 or degree 3).

Let Gv1 denote the graph constructed from switching of vertex v1 of G.

Corresponding to the vertices of different degree in Cn, it is required to discuss


Case 1: d(v1) = 2.

Then by the effect of switching operation, the edge set of Gv1 is

E(Gv1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v2vn}⋃{v1vj | 3 ≤ j ≤ n− 1}.

It is to be noted that, |V (Gv1)| = n and |E(Gv1)| = 2n− 4.

Consider a bijection f : V (Gv1)→ {1, 3, 5, . . . , 2n− 1} defined as below.

f(vj) =

2j − 1; 1 ≤ j ≤ 4.

2j + 1; 5 ≤ j ≤ n− 1.

f(vn) = 9.

If label of vn−1 is a multiple of 9 then interchange the labels of vn−1 and vn−2.


ef (1) = n− 2 = ef (0).


Case 2: d(v1) = 3.


E(Gv1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v1vj | 3 ≤ j ≤ n− 2}.It is to be noted that, |V (Gv1)| = n and |E(Gv1)| = 2n− 6.

62


Consider a bijection f : V (Gv1)→ {1, 3, 5, . . . , 2n− 1} defined as below.

f(vj) =

2j − 1; j = 1, 2, 6 ≤ j ≤ n.

2j − 3; 4 ≤ j ≤ 5.

f(v3) = 9.


ef (1) = n− 3 = ef (0).


That is, the graph Gv, where G is cycle Cn with one chord, is VODC.

Example 4.6.1. VODC labeling in (G)v, where d(v) = 2 and 3, G = C7 with one

chord, are demonstrated in the following Figure 4.11.

1

5

1

7

3

5v6

v5v4

v3

v2

v1

3

11

9 9

11

1313

7

v7

Figure 4.11: VODC labeling in (G)v.

Theorem 4.6.2. Gv is VODC, where G is cycle with twin chords Cn,3.

Proof. Let V (Cn,3) = {vj | 1 ≤ j ≤ n} = V (Cn).

Let E(Cn,3) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{v2vn}

⋃{v2vn−1}, where v2vn,

v2vn−1 are chords.

WLOG let v1 be the switched vertex.

Let (Cn,3)v1 denote the graph constructed from switching of vertex v1 of Cn,3.

Corresponding to the vertices of different degree in Cn,3, it is required to discuss

following three cases.

Case 1: d(v1) = 2.

63


Then by the effect of switching operation, the edge set of (Cn,3)v1 is

E((Cn,3)v1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v2vn}⋃{v2vn−1}

⋃{v1vj | 3 ≤ j ≤ n− 1}.It is to be noted that, |V ((Cn,3)v1)| = n and |E((Cn,3)v1)| = 2n− 3.

We define labeling function f : V ((Cn,3)v1)→ {1, 3, 5, . . . , 2n− 1} as below.

f(vj) =

2j − 1; 1 ≤ j ≤ 4.

2j + 1; 5 ≤ j ≤ n− 1.

f(vn) = 9.



ef (1) = n− 2, ef (0) = n− 1.


Case 2: d(v1) = 3.


E((Cn,3)v1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v3vn}⋃{v1vj | 4 ≤ j ≤ n− 1}.

It is to be noted that, |V ((Cn,3)v1)| = n and |E((Cn,3)v1)| = 2n− 5.

Consider a bijection f : V ((Cn,3)v1)→ {1, 3, 5, . . . , 2n− 1} defined as below.

f(vj) =

2j − 1; j = 1, 2, 6 ≤ j ≤ n.

2j − 3; 4 ≤ j ≤ 5.

f(v3) = 9.



ef (1) =⌊2n−5

2

⌋, ef (0) =

⌈2n−5

2

⌉


Case 3: d(v1) = 4.


E((Cn,3)v1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v1vj | 3 ≤ j ≤ n− 3}.

64


It is to be noted that, |V ((Cn,3)v1)| = n and |E((Cn,3)v1)| = 2n− 7.

Consider a bijection f : V ((Cn,3)v1)→ {1, 3, 5, . . . , 2n− 1} defined as below.

f(vj) =

2j − 1; j = 1, 2, 6 ≤ j ≤ n.

2j − 3; 4 ≤ j ≤ 5.

f(v3) = 9.


ef (1) =⌊2n−7

2

⌋, ef (0) =

⌈2n−7

2

⌉.


Hence, the graph under consideration admits VODC labeling in each case.

That is, (Cn,3)v is VODC graph.

Example 4.6.2. C8,3 graph and VODC labeling in (C8,3)v, where d(v) = 2, 3, 4 are

demonstrated in the following Figure 4.12.

v4 v6

11

13

15 11

13

151

5

7

31

5

7

3

v3

v5

v7

v1

v2 1

3

7

511

9

9

9

13

15

v8

Figure 4.12: VODC labeling in (C8,3)v

Theorem 4.6.3. (Cn(1, 1, n− 5))v is VODC.

Proof. Let V (Cn(1, 1, n− 5)) = {vj | 1 ≤ j ≤ n} = V (Cn)

Let E(Cn(1, 1, n − 5)) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{v1v3}⋃{v3vn−1}

⋃{vn−1v1},where v1vn−1, v1v3, vn−1v3 are chords of Cn.


Let (Cn(1, 1, n− 5))v1 denote the graph constructed from switching of a vertex v1 of

Cn(1, 1, n− 5).

Corresponding to the vertices of different degree in Cn(1, 1, n− 5), it is required to

discuss following two cases.

65


Case 1: d(v1) = 2.

Then by the effect of switching operation, the edge set of (Cn(1, 1, n− 5))v1 is

E((Cn(1, 1, n − 5))v1) = {vjvj+1 | 2 ≤ j ≤ n − 1}⋃{v2v4}⋃{v4vn−1}

⋃{vn−1v2}⋃{v1vj | 3 ≤ j ≤ n− 1}.It is to be noted that, |V ((Cn(1, 1, n−5))v1)| = n and |E((Cn(1, 1, n−5))v1)| = 2n−2.

Let us a function f from V ((Cn(1, 1, n−5))v1) to {1, 3, 5, . . . , 2n−1} defined as below.

f(vj) =

2j − 3; j = 2, 3, 7 ≤ j ≤ 8.

2j − 5; 5 ≤ j ≤ 6.

2j − 1; 9 ≤ j ≤ n− 2.

f(v4) = 9.

f(vn−1) = 15.

f(vn) = p2;

f(v1) = p1;

where p1 = max{x | x is a prime number and x ≤ 2n− 1} and

p2 = max{y | y is a prime number and y < p1}.By looking into the above prescribed pattern,

ef (1) = n− 1 = ef (0).


Case 2: d(v1) = 4.

Then by the effect of switching operation, the edge set of (Cn(1, 1, n− 5))v1 is

E((Cn(1, 1, n−5))v1) = {vjvj+1 | 2 ≤ j ≤ n−1}⋃{v3vn−1}⋃{v1vj | 4 ≤ j ≤ n−2}.

It is to be noted that, |V ((Cn(1, 1, n−5))v1)| = n and |E((Cn(1, 1, n−5))v1)| = 2n−6.

66

4.7. VODC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs

Consider a bijection f : V ((Cn(1, 1, n−5))v1)→ {1, 3, 5, . . . , 2n−1} defined as below.

f(vj) =

2j − 1; j = 1, 2, 7 ≤ j ≤ 8.

2j − 3; 4 ≤ j ≤ 6.

2j + 1; 9 ≤ j ≤ n− 2.

f(v3) = 9.

f(vn−1) = 15.

f(vn) = p, where p = max {x | x is a prime number and x ≤ 2n− 1}.


ef (1) = n− 3 = ef (0).


That is, (Cn(1, 1, n− 5))v is VODC.

Example 4.6.3. C8(1, 1, 3) graph and VODC labeling in (C8(1, 1, 3))v, where d(v) =

2, 4 are demonstrated in the following Figure 4.13.

v3

v5

v7

v8v2

v1 1

3

9

5

7

11

15

13 13

9

57

13

11

15

v4 v6

Figure 4.13: VODC labeling in (C8(1, 1, 3))v

4.7 VODC Labeling With the Use of Switching of a Vertex

in Wheel and Shell Allied Graphs

Theorem 4.7.1. (Wn)v is VODC, where v is rim vertex.

67


Proof. Let V (Wn) = {vj | 0 ≤ j ≤ n}, where v0 is the apex vertex and vj(1 ≤ j ≤ n)

are the rim vertices of wheel Wn.

Let E(Wn) = {vjvj+1 | 1 ≤ j ≤ n− 1}⋃{vnv1}⋃{v0vj | 1 ≤ j ≤ n}.

Let (Wn)v1 denote the graph constructed from switching of a rim vertex v1 of Wn.

Then by the effect of switching operation, the edge set of (Wn)v1 is

E((Wn)v1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v0vj | 2 ≤ j ≤ n}⋃{v1vj | 3 ≤ j ≤ n− 1}.It is to be noted that, |V ((Wn)v1)| = n+ 1 and |E((Wn)v1)| = 3n− 6.

Let us a function f : V ((Wn)v1))→ {1, 3, 5, . . . , 2n+ 1} defined as below.

Case 1: n ≤ 9.

f(v0) = 1.

f(vj) = 2j + 1; 1 ≤ j ≤ n.

Case 2: n > 9.

f(v0) = 1.

f(v1) = 3.

f(v2j) = pj+2; 1 ≤ j ≤ k,

f(v2j+1) = 3f(v2j); 1 ≤ j ≤ k,

where pj+2 = (j + 2)th prime number and k =⌊n−66

⌋.



n ≡ 0, 2(mod 4) ef (1) = 3n−62

= ef (0)

n ≡ 1, 3(mod 4) ef (1) =⌈3n−6

2

⌉, ef (0) =

⌊3n−6

2

⌋

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, (Wn)v is VODC, where v is rim vertex.

Example 4.7.1. Wheel graph W9 and VODC labeling in (W9)v1, where v1 is rim

vertex are demonstrated in the following Figure 4.14.

68


v1

v2

v3

v4

v5 v6

v7

v8

v9

v0 1

53

9

7

11

19

17

15

13

Figure 4.14: VODC labeling in (W9)v1

Theorem 4.7.2. Gv is VODC, where G is gear graph Gn and v is not apex vertex.

Proof. Let V (Gn) = {v0}⋃{vj | 1 ≤ j ≤ 2n}, where d(v0) = n and vj(1 ≤ j ≤ 2n)

are other vertices of gear graph Gn,

deg(vj) =

2 when j is even;

3 when j is odd; 1 ≤ j ≤ 2n.

Let E(Gn) = {u0u2j−1 | 1 ≤ j ≤ n}⋃{ujuj+1 | 1 ≤ j ≤ 2n− 1}⋃{u2nu1}.(Gn)ui

∼= (Gn)uj , where d(ui) = d(uj).

Let (Gn)ui denote the graph constructed from switching of vertex uj (j = 1, 2) of

Gn.



Case 1: deg(u1) = 3.


E((Gn)u1) = {u0u2j−1 | 2 ≤ j ≤ n}⋃{ujuj+1 | 2 ≤ j ≤ 2n − 1}⋃{u1uj | 3 ≤ j ≤2n− 1}.It is to be noted that, |V (Gn)u1| = 2n+ 1 and |E(Gn)u1| = 5n− 6.

Consider a bijection f : V ((Gn)u1)→ {1, 3, . . . , 4n+ 1} defined as below.

Let f(u1) = 1

69


which will generate 2n− 3 edges (which are adjacent with v1) with label 1.

Let f(u0) = 3.

Now it remains to generate k = 5n−62− (2n− 3) edges with label 1.

For the vertices u3, u5, . . . , uk, allocate the vertex labels as per following ordered


f(u2j+1) = 3(2j + 1); 1 ≤ j ≤ k,

where k =

n2; if n is even.

n+12

; if n is odd.

Allocate labels {5, 7, 11, . . . , 2n} to the vertices {u2, u4, . . .} and {uk+1, uk+2, . . . , u2n}in any order.



n is even ef (1) = 5n−62

= ef (0)

n is odd ef (1) = 5n−52, ef (0) = 5n−7

2




E((Gn)u2) = {u0u2j−1 | 1 ≤ j ≤ n}⋃{ujuj+1 | 3 ≤ j ≤ 2n − 1}⋃{u2nu1}⋃{u2uj |

4 ≤ j ≤ 2n}.It is to be noted that, |V (Gn))u2 | = 2n+ 1 and |E(Gn))u2| = 5n− 4.

Consider a bijection f : V ((Gn)u2)→ {1, 3, . . . , 4n+ 1} defined as below.

Let f(u1) = 1

which will generate 2n− 3 edges (which are adjacent with v1) with label 1.

Let f(u0) = 3.

Now it remains to generate k = 5n−62− (2n− 3) edges with label 1.

70


For the vertices u2, u4, . . . , uk, allocate the vertex labels as per following ordered


f(u2j) = 3(2j + 1); 1 ≤ j ≤ k,

where k =

n2; if n is even.

n+12

; if n is odd.

Allocate labels {5, 7, 11, . . . , 2n} to the vertices {u3, u5, . . .} and {uk+1, uk+2, . . . , u2n}in any order.



n is even ef (1) = 5n−42

= ef (0)

n is odd ef (1) = 5n−32, ef (0) = 5n−5

2


Hence, the graph under consideration admits VODC labeling.

That is, (Gn)v (v is not apex vertex) is VODC.

Example 4.7.2. VODC labeling in (G6)v, where d(v) = 3, d(v) = 2 are demon-


3

13

7

1

59

3

11

7

913

1

15 17

17

19

19

21

21

15

23

2523

5

11

u12 25

u0

u2

u3

u4

u5

u6

u7

u8

u9

u10

u1

u11

Figure 4.15: VODC labeling in (G6)v

Theorem 4.7.3. Gv is VODC, where G is shell graph Sn and v is not apex vertex.

Proof. Let V (Sn) = {uj | 0 ≤ j ≤ n − 1}, where u0 is the apex vertex and uj(1 ≤j ≤ n− 1) are the other vertices of shell Sn, where

71


deg(uj) =

2 when j = 1, n− 1.

3 when 2 ≤ j ≤ n− 2.

Let E(Sn) = {u0uj | 2 ≤ j ≤ n− 2}⋃{ujuj+1 | 0 ≤ j ≤ n− 2}⋃{u0un−1}.(Sn)ui

∼= (Sn)uj , where d(ui) = d(uj).

Let (Sn)ui denote the graph constructed from switching of vertex uj (j = 1, 2) of

Sn.

Corresponding to the vertices of different degree in Sn, it is required to discuss fol-

lowing two cases.

Case 1: d(u2) = 3.


E((Sn)u2) = {u0uj | 3 ≤ j ≤ n−2}⋃{ujuj+1 | 3 ≤ j ≤ n−1}⋃{un−1u0}{u0u1}⋃{u2uj |

4 ≤ j ≤ n− 1}.It is to be noted that, |V ((Sn)u2)| = n and |E((Sn)u2)| = 3n− 10.

Consider a bijection f : V ((Sn)u2)→ {1, 3, . . . 2n− 1} as per the following subcases.

Subcase 1: n ≤ 15.

f(u0) = 1.

f(u1) = p,where p = max {x | x is a prime number and x ≤ 2n− 1}.

f(u2) = 3.

f(uj) = 2j − 1; 3 ≤ j ≤ n− 1.

Subcase 2: n > 15.

f(u0) = 1.

f(u1) = p,where p = max {x | x is a prime number and x ≤ 2n− 1}.

f(u2) = 3.

f(u2j−1) = pj; 2 ≤ j ≤ k,

f(u2j) = 3f(u2j−1); 2 ≤ j ≤ k,

where pj denotes the jth prime number and k =⌊n−10

6

⌋.


72



n ≡ 0, 2(mod 4) ef (1) = 3n−102

= ef (0)

n ≡ 1, 3(mod 4) ef (1) = 3n−92, ef (0) = 3n−11

2


Case 2: d(u1) = 2.


E((Sn)u1) = {u0uj | 2 ≤ j ≤ n − 1}⋃{ujuj+1 | 2 ≤ j ≤ n − 1}⋃{un−1u0}{u1uj |3 ≤ j ≤ n− 1}.It is to be noted that, |V ((Sn)u1)| = n, |E((Sn)u1)| = 3n− 8.

Consider a bijection f : V ((Sn)u1)→ {1, 3, . . . 2n− 1} defined as below.

Subcase 1: n ≤ 9.

f(uj) = 2j + 1; 0 ≤ j ≤ n− 1.

Subcase 2: n > 9.

f(u0) = 1.

f(u1) = 3.

f(u2j) = pj+2; 1 ≤ j ≤ k,

f(u2j+1) = 3f(u2j); 1 ≤ j ≤ k,

where pj+2 is (j + 2)th prime number and k =⌊n−46

⌋.



n ≡ 0, 2(mod 4) ef (1) = 3n−82

= ef (0)

n ≡ 1, 3(mod 4) ef (1) = 3n−72, ef (0) = 3n−9

2


That is, (Sn)v is VODC and v is not apex vertex.

Example 4.7.3. Shell graph S7 and VODC labeling in (S7)v, where d(v) = 3, d(v) =

2 are demonstrated in the following Figure 4.16.

73


1

3

5

u2

u3 u4

u5

u1

u0

u6

1

3

5

7

9

9

13

7

13 11

11

Figure 4.16: VODC labeling in (S7)v

Theorem 4.7.4. (Fln)v is VODC, where v is not apex vertex.

Proof. Let V (Fln) = {u0}⋃{uj | 1 ≤ j ≤ n}⋃{vj | 1 ≤ j ≤ n}, where u0 is the

apex vertex, uj(1 ≤ j ≤ n) are the internal vertices and vj(1 ≤ j ≤ n) are the

external vertices; deg(ui) = 4 and deg(vi) = 2.

Let E(Fln) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{unu1}⋃{u0uj | 1 ≤ j ≤ n}⋃{u0vj | 1 ≤

j ≤ n}⋃{ujvj | 1 ≤ j ≤ n}, where ujvj are the spoke edges.

(Fln)ui∼= (Fln)uj , where d(ui) = d(uj).





Case 1: d(u1) = 4.

By the effect of switching operation, the edge set of (Fln)u1 is

E((Fln)u1) = {ujuj+1 | 2 ≤ j ≤ n}⋃{u0uj | 2 ≤ j ≤ n}⋃{u0vj | 1 ≤ j ≤n}⋃{ujvj | 2 ≤ j ≤ n}⋃{u1uj | 3 ≤ j ≤ n− 1}⋃{u1vj | 2 ≤ j ≤ n}.It is to be noted that, |V ((Fln)u1)| = 2n+ 1, |E((Fln)u1)| = 6n− 8.

Consider a bijection f : V ((Fln)u1)→ {1, 3, . . . 4n+ 1} defined as below.

Subcase 1: n ≤ 8.

f(u0) = 1.

f(u1) = 3.

f(v1) = p,where p = max {x | x is a prime number and x ≤ 4n+ 1}.

74


For the vertices u2, u3, . . . , un, v2, v3, . . . , vn of (Fln)u1 allocate the vertex labels such

that for any edge ujuj+1 ∈ E(Fln)u1 ,

f(uj) - f(uj+1)

f(uj) - f(vj); 2 ≤ j ≤ n.

Subcase 2: n > 8.

f(u0) = 1.

f(u1) = 3.

f(v1) = p,where p = max {x | x is a prime number and x ≤ 4n+ 1}.

f(uj) = pj; 2 ≤ j ≤ k,

f(vj) = 3f(uj); 2 ≤ j ≤ k,

where pj is jth prime number and k =⌊n3

⌋− 2.

For the vertices uk+1, uk+2, . . . , un, vk+1, vk+2, . . . , vn of (Fln)u1 , allocate the vertex

labels such that for any edge ujuj+1 ∈ E(Fln)u1 ,

f(uj) - f(uj+1); k + 1 ≤ j ≤ n− 1

f(uj) - f(vj); k + 1 ≤ j ≤ n.


ef (1) = 3n− 4 = ef (0).


Case 2: d(v1) = 2.


E((Fln)v1) = {ujuj+1 | 1 ≤ j ≤ n}⋃{u0uj | 1 ≤ j ≤ n}⋃{u0vj | 2 ≤ j ≤n}⋃{ujvj | 2 ≤ j ≤ n}⋃{v1uj | 2 ≤ j ≤ n− 1}⋃{v1vj | 2 ≤ j ≤ n}.It is to be noted that, |V ((Fln)v1)| = 2n+ 1, |E((Fln)v1)| = 6n− 4.

75


Consider a bijection f : V ((Fln)v1)→ {1, 3, . . . 4n+ 1} defined as below.

f(u0) = 1.

f(v1) = 3.

f(u1) = p,where p = max {x | x is a prime number and x ≤ 4n+ 1}.

f(uj) = pj+1; 2 ≤ j ≤ k,

f(vj) = 3f(uj); 2 ≤ j ≤ k,

where pj+1 = (j + 1)th prime number and k =⌊n3

⌋.

Allocate the labels to vertices uk+1, uk+2, . . . , un, vk+1, vk+2, . . . , vn of (Fln)v1 such

that

f(uj) - f(uj+1); k + 1 ≤ j ≤ n− 1

f(uj) - f(vj); k + 1 ≤ j ≤ n.


ef (1) = 3n− 2 = ef (0).


That is, (Fln)v is VODC (v is not apex vertex) .

Example 4.7.4. Flower graph Fl4 and VODC labeling in (Fl4)v, where d(v) = 2

and 4 are demonstrated in the following Figure 4.17.

3

9

13

11

75

17

15

1

u4

v2v3

u2 u3

u1

v1v4

u01

9

7

3

5

11 13

1715

Figure 4.17: VODC labeling in (Fl4)v

76

4.8. VODC Labeling With the Use of Ringsum of Different Graphs with Star Graph K1,n

4.8 VODC Labeling With the Use of Ringsum of Different

Graphs with Star Graph K1,n

Ghodasara and Rokad[5] illumined and derived some captivating results on cordial


Under the inspiration of this credibility, in current segment we demonstrate few

graphs constructed from the graph operation ringsum for VODC labeling.

Theorem 4.8.1. G⊕K1,n is VODC graph, where G is cycle Cn with one chord.

Proof. Let G denote the cycle Cn with one chord.

Let V (G ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (G) = V (Cn) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.


where u2un is the chord of Cn and the egdes u1u2, u2un, u1un form a triangle.


Consider a bijection f : V (G⊕K1,n)→ {1, 3, . . . , 4n− 1} defined as below.

f(u1) = 3.

f(u2) = 1.

f(vj) = 3(2j + 1); 1 ≤ j ≤ k1, where k1 =

⌊4n− 3

6

⌋.

f(u2j+1) = pj+2; 1 ≤ j ≤ k2,

f(u2j+2) = 5f(u2j+1); 1 ≤ j ≤ k2,

where pj+2 is (j + 2)th prime number and k2 =⌊n−53

⌋.

For the vertices u2k2+3, u2k2+4, . . . , un and vk1+1, vk1+2, . . . , vn allocate the vertex la-

bels such that for any edge ujuj+1 ∈ E(G⊕K1,n),

f(uj) - f(ui+1), 2k2 + 3 ≤ i ≤ n;

f(u1) - f(vi), k + 1 ≤ i ≤ n.

77



ef (0) = n+ 1, ef (1) = n.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, G⊕K1,n is a VODC graph, where G is the cycle Cn with one chord.

Example 4.8.1. VODC labeling in ringsum of C7 with one chord and K1,7 is demon-

strated in Figure 4.18.

v3 v4 v5 v6 v7v1 v2

3

5

7

9

11

u3

u4 u5

u6

u7

u1

u2 117

15 1921 27 2523

13

Figure 4.18: VODC labeling in the graph constructed from ringsum of C7 with one chord and K1,7

Theorem 4.8.2. Cn,3 ⊕K1,n is a VODC graph.



n}⋃{u2un}⋃{u2un−1}, where unu2 and u2un−1 are the chords.


Consider a bijection f : V (Cn,3 ⊕K1,n)→ {1, 3, . . . , 4n− 1} defined as below.

f(u1) = 3.

f(u2) = 1.

f(vj) = 3(2j + 1); 1 ≤ j ≤ k1, k1 =

⌊4n− 3

6

⌋.

f(u2j+1) = pj+2; 1 ≤ i ≤ k2,

f(u2j+2) = 5f(u2j+1); 1 ≤ j ≤ k2,

78



⌋.


bels such that for any edge e = ujuj+1 ∈ E(Cn,3 ⊕K1,n),

f(uj) - f(uj+1), 2k2 + 3 ≤ j ≤ n.

f(u1) - f(vj), k1 + 1 ≤ j ≤ n.


ef (0) = n+ 1 = ef (1).

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Cn,3 ⊕K1,n is a VODC graph.

Example 4.8.2. VODC labeling in C8,3⊕K1,8 is demonstrated in the Figure 4.19.

v3 v6v1

u3

u6

u7

v8

u2 u81

5

7

9

11

13

15

3

v5 v7

u5

u4

u1

25

17

21 27 19 23 29` 31

v4v2

Figure 4.19: VODC labeling in C8,3 ⊕K1,8

Theorem 4.8.3. Cn(1, 1, n− 5)⊕K1,n is a VODC graph.

Proof. Let G denote cycle with triangle Cn(1, 1, n− 5).



Let E(G ⊕ K1,n) = {ujuj+1 | 1 ≤ j ≤ n − 1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤

n}⋃{u1u3}⋃ {u3un−1}

⋃{un−1u1}, where u1, u3 and un−1 are the vertices of tri-

angle formed by the chords u1u3, u3un−1 and u1un−1.

79



Consider a bijection f : V (G⊕K1,n)→ {1, 3, . . . , 4n− 1} defined as below.

f(u1) = 3.

f(u3) = 1.

f(u2) = p,where p = max {x | x is the largest prime number x ≤ 4n− 1},

f(vj) = 3(2j + 1); 1 ≤ j ≤ k1, where k1 =

⌊4n− 3

6

⌋.

f(u2(j+1)) = pj+2; 1 ≤ j ≤ k2,

f(u2j+3) = 5f(u2(j+1)); 1 ≤ j ≤ k2,


⌋.


bels such that for any edge ujuj+1 ∈ E(G⊕K1,n),

f(uj) - f(uj+1), 2k2 + 4 ≤ j ≤ n.

f(v) - f(vj), k1 + 1 ≤ j ≤ n.


ef (0) = n+ 1, ef (1) = n+ 2.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Cn(1, 1, n− 5)⊕K1,n is a VODC graph.

Example 4.8.3. VODC labeling in the graph C8(1, 1, 3) ⊕K1,8 is demonstrated in

the following Figure 4.20.

80


v3v1

u3

u4

u6

u7

u2

u1

u5

u8

v2 v4 v5 v6 v7 v8

13

159

11

5 7

1

3

25

21 27 17 19 23 29

31

Figure 4.20: VODC labeling in C8(1, 1, 3)⊕K1,8

Theorem 4.8.4. Pn ⊕K1,n is a VODC graph.



Consider a bijection f : V (Pn ⊕K1,n)→ {1, 3, 5, . . . , 4n− 1} defined as below.

f(u1) = 3.

f(u2) = 1.

f(vj) = 3(2j + 1); 1 ≤ j ≤ k1, where k1 =

⌊4n− 3

6

⌋.

f(u2j+1) = pj+2; 1 ≤ j ≤ k2,

f(u2j+2) = 5f(u2j+1); 1 ≤ j ≤ k2,


⌋.


bels such that for any edge e = ujuj+1 ∈ E(Pn ⊕K1,n),

f(ui) - f(ui+1), 2k2 + 3 ≤ i ≤ n.

f(v) - f(vj), k1 + 1 ≤ j ≤ n.

81



ef (1) = n− 1, ef (0) = n.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Pn ⊕K1,n is a VODC graph.

Example 4.8.4. VODC labeling in the graph P5 ⊕ K1,5 is demonstrated in the


u1u2u3u4u5

v2

v3

v4

57

v1

v5

31

9

11

15

13

1719

Figure 4.21: VODC labeling in P5 ⊕K1,5

Theorem 4.8.5. Wn ⊕K1,n is a VODC graph.

Proof. Let V (Wn ⊕K1,n) = {u, uj, vj | 1 ≤ j ≤ n}, where V (Wn) = {u, uj | 1 ≤ j ≤n} and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here u is the apex vertex, ui(1 ≤ i ≤ n) are rim vertices of Wn and vi(1 ≤ i ≤ n)

are the pendant vertices, u1 is apex vertex of star graph.

Let E(Wn⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n−1}⋃{unu1}⋃{uuj | 1 ≤ j ≤ n}⋃{u1vj |


Consider a bijection f : V (Wn ⊕K1,n)→ {1, 3, 5, . . . , 4n+ 1} defined as below.

f(u1) = 3.

f(u) = 1.

f(vj) = 3(2j + 1); 1 ≤ j ≤ k,

f(uj+1) = pj+2; 1 ≤ j ≤ k,

where pj+2 is (j + 2)th prime number and k =⌊n2

⌋.

For the vertices uk+2, uk+3, . . . , un, vk+1, vk+2, . . . , vn allocate the vertex labels such

82


that for any edge ujuj+1 ∈ E(Wn ⊕K1,n),

f(uj) - f(uj+1), k + 2 ≤ j ≤ n.

f(u1) - f(vj), k + 1 ≤ j ≤ n.


ef (1) =

⌈3n

2

⌉, ef (0) =

⌊3n

2

⌋.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Wn ⊕K1,n is a VODC graph.

Example 4.8.5. VODC labeling in the graph W6 ⊕ K1,6 is demonstrated in the


u

v6

u3

u4

u5

u1

u2

1

3

5

7

9

11

13

17

15 21 19 2523

v1 v2 v4v3 v5

u6

Figure 4.22: VODC labeling in W6 ⊕K1,6

Theorem 4.8.6. Fln ⊕K1,n is a VODC graph.

Proof. Let V (Fln ⊕K1,n) = {u, uj, vj, wj | 1 ≤ j ≤ n}, where V (Fln) = {u, uj, wj |1 ≤ j ≤ n} and V (K1,n) = {w1, vj | 1 ≤ j ≤ n}.Here u is apex vertex, uj(1 ≤ j ≤ n) are internal vertices and wj(1 ≤ j ≤ n) are

external vertices of Fln; d(vj) = 1(1 ≤ j ≤ n), w1 is apex vertex of star graph.

Let E(Fln⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n−1}⋃{unu1}⋃{uuj | 1 ≤ j ≤ n}⋃{uwj |

1 ≤ j ≤ n}⋃{ujwj | 1 ≤ j ≤ n}⋃{w1vj | 1 ≤ j ≤ n}.It is to be noted that, |V (Fln ⊕K1,n)| = 3n+ 1 and |E(Fln ⊕K1,n)| = 5n.

83


Consider a bijection f : V (Fln ⊕K1,n)→ {1, 3, 5, . . . , 6n+ 1} defined as below.

f(u) = 1.

f(w1) = 3.

f(vj) = 3(2j + 1); 1 ≤ j ≤ k, where k =⌈n

2

⌉.

For the vertices vk+1, vk+2, . . . , vn and u1, u2 . . . un, w2, w3 . . . wn allocate the vertex

labels such that for any edge ujuj+1 ∈ E(Fln ⊕K1,n),

f(w1) - f(vj); k + 1 ≤ j ≤ n,

f(uj) - f(uj+1), f(uj) - f(wj); 1 ≤ j ≤ n.


ef (1) =

⌈5n

2

⌉, ef (0) =

⌊5n

2

⌋.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Fln ⊕K1,n is a VODC.

Example 4.8.6. VODC labeling in the graph Fl4 ⊕ K1,4 is demonstrated in the


u1

u2

u3

u4

w1

v1

v2 v3

v4

w4

w2

w3

u1

3

79

5

1113

15

17

19

21 23

25

Figure 4.23: VODC labeling in Fl4 ⊕K1,4

Theorem 4.8.7. K2,n ⊕K1,n is a VODC graph.

84



Let E(K2,n ⊕K1,n) = {uuj | 1 ≤ j ≤ n}⋃{wuj | 1 ≤ j ≤ n}⋃{u1vj | 1 ≤ j ≤ n}.It is to be noted that, |V (K2,n ⊕K1,n)| = 2n+ 2 and |E(K2,n ⊕K1,n)| = 3n.

Consider a bijection f : V (K2,n ⊕K1,n)→ {1, 3, 5, . . . , 4n+ 3} defined as below.

f(u) = 1.

f(w) = p.

f(u1) = 3.

f(vj) = 3(2j + 1); 1 ≤ j ≤ k,

f(uj+1) = pj+2; 1 ≤ j ≤ n− 1,

where, p = max {x | x is the largest prime number x ≤ 4n+ 3},pj+2 is (j + 2)th prime number and k =

⌈n2

⌉.

For the vertices vk+1, vk+2, . . . , vn allocate the vertex labels s. t.

f(v) - f(vj), k + 1 ≤ j ≤ n.



n ≡ 0, 2(mod 4) ef (1) = 3n2

= ef (0)

n ≡ 1, 3(mod 4) ef (1) =⌈3n2

⌉, ef (0) =

⌊3n2

⌋

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, K2,n ⊕K1,n is a VODC graph.

Example 4.8.7. VODC labeling in the graph K2,7 ⊕ K1,7 is demonstrated in the


85


17

19

29

u1

u2

v1

v2

u3

v3 v4

v5

u4

w

u

u5

u6

u7

v6

v7

1

31

5

7

11

13

9

1521 27

25

23

3

Figure 4.24: VODC labeling in K2,7 ⊕K1,7

Theorem 4.8.8. DFn ⊕K1,n is a VODC graph.

Proof. Let V (DFn ⊕K1,n) = {u,w, uj, vj | 1 ≤ j ≤ n}, where V (DFn) = {u,w, uj |1 ≤ j ≤ n} and V (K1,n) = {u, vj | 1 ≤ j ≤ n}.Here u,w are apex vertices of DFn, u1, u2, . . . , un are vertices of path Pn correspond-

ing to DFn, d(vj) = 1, where 1 ≤ j ≤ n and u is apex vertex of star graph.


Consider a bijection f : V (DFn ⊕K1,n)→ {1, 3, 5, . . . , 4n+ 3} defined as below.

f(w) = 3.

f(u) = 1.

f(uj) = pj+2; 1 ≤ j ≤ n,

f(vj) = 3(2j + 1); 1 ≤ j ≤ k,

where pj+2 is (j + 2)th prime number and k =⌊n2

⌋.

For the vertices vk+1, vk+2, . . . , vn allocate the vertex labels such that

f(uj) - f(uj+1), 2k + 1 ≤ j ≤ n.

86



ef (1) = 2n, ef (0) = 2n− 1.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, DFn ⊕K1,n is a VODC graph.

Example 4.8.8. VODC labeling in the graph DF5 ⊕ K1,5 is demonstrated in the


u1 u2 u3 u4 u5

9v1

v2

v3

v4

v5

u

w

5 11 13 17

19

23

3

1

15

7

21

Figure 4.25: VODC labeling in DF5 ⊕K1,5


In this chapter we have emanated some new square DC, CDC and VODC graphs. We

have derived new results on these labelings using graph operations such that one

point union, arbitrary supersubdivision and duplication. We have also emanated

VODC labeling for larger graphs which are constructed from a standard graph by

means of graph operations such as ringsum and switching of a vertex.

However, there is no such standard relation between either of the three Square DC,

CDC and VODC labelings; like, we may find a graph which admits one of the three

labelings but not the other. To observe this matter more effectively, the list of graph

families satisfying/ not satisfying certain labeling is shown in following table.

87


Graph Square DC CDC VODC

Wn Yes Yes Yes

P7 Yes No No

K4 No Yes No

Wn ⊕K1,n No No Yes

K1,n(n ≥ 8) No No No

Still there is a big scope of research and extension/generalization of the results de-

rived here. Here we propose some open problems which may provide better direction

for further delevopment in these labelings.


be a VODC graph.

Problem 4.9.2. Investigate some new square DC, CDC, VODC graphs with respect

to other graph operations.

Problem 4.9.3. Classify/Generalize the graph G such that G ⊕ K1,n is a square

DC, CDC, VODC graph. (Here |V (G)| may or may not be equal to n.)

In the next chapter, we will discuss sum DC labeling of graphs.

88

CHAPTER 5

Sum Divisor Cordial Labeling of

Graphs

Different variants of DC labeling namely square DC, CDC and VODC labeling were

discussed in the earlier Chapter-4, while the current chapter aims to deliberate

another labeling having DC theme.

Inspired by the idea of DC labeling in 2016, A. Lourdusamy and F. Patrick originated

the concept of one of the variant of DC labeling called sum DC labeling.

5.1 Introduction

Definition 5.1.1 (Lourdusamy et al.[1]). Let G = (V,E) be a simple graph with


edge labeling function f ∗ : E(G)→ {0, 1} be defined as

f ∗(e = uv) =

1; if 2 | [f(u) + f(v)]

0; otherwise

Then the function f is called a sum DC labeling if |ef (0)− ef (1)| ≤ 1.

A graph which confesses SDC labeling is called SDC graph.

Example 5.1.1. A SDC labeling of star graph K1,7 is demonstrated in the following

Figure 5.1.

89

5.2. Some Existing Results on Sum DC Labeling

1

2

3

4 6

5

8

7v0

v1

v2

v3

v4

v5

v6

v7

Figure 5.1: SDC labeling in K1,7

It is clear that for a graph, we may find more than one SDC labeling by using proper

permutation of the labels used (such that the condition for edge labels is satisfied).

5.2 Some Existing Results on Sum DC Labeling

SDC labeling was introduced in 2016. Since then, many researchers have explored

this labeling by finding captivating results during last three years.

In first paper on SDC labeling[1], Lourdusamy et al. have established SDC labeling

for some basic graphs and derived the following results.

Theorem 5.2.1. The path Pn is SDC.

Theorem 5.2.2. The comb Pn �K1 and the crown Cn �K1 are SDC graph.

Theorem 5.2.3. The star graph K1,n and the barycentric subdivision of the star

K1,n are SDC.

Theorem 5.2.4. The complete bipartite graph K2,n is SDC.

Theorem 5.2.5. The graph K2 +mK1 is SDC graph.

Theorem 5.2.6. The bistar Bn,n and restricted square of bistar Bn,n are SDC.

Theorem 5.2.7. The flower graph Fln, gear graph Gn and the jewel graph Jn are

SDC.

In another research paper[2], the same authors have derived more results in which

characterize a SDC graph. They have also developed SDC labeling in some K1,n

and Bm,n allied graphs. These results are listed below.

90

5.3. Some New Cycle Related Sum DC Graphs

Theorem 5.2.8. The barycentric subdivision of bistar Bn,n is SDC.

Theorem 5.2.9. S ′(K1,n) and D2(K1,n) of star K1,n are SDC.

Theorem 5.2.10. S ′(Bn,n) and DS(Bn,n) of bistar Bn,n are SDC.

Theorem 5.2.11. D2(Bn,n) is SDC.

Theorem 5.2.12. The closed helm graph CHn is SDC graph.

Rozario and Surya[38] have discussed SDC labeling for switching invariance in path

Pn and cycle Cn. These results are listed below.

Theorem 5.2.13. For n ∈ N, there is a SDC graph G which has n vertices.

Theorem 5.2.14. Switching invarivance in path Pn is SDC.

Theorem 5.2.15. Switching invarivance in cycle Cn is SDC.

5.3 Some New Cycle Related Sum DC Graphs

A. Lourdusamy and F. Patrick[1] demonstrate some fascinating graphs for SDC

labeling. Under the motivation of this belief, in current section we originate SDC

labeling of some cycle related graphs.

Theorem 5.3.1. Cycle Cn is SDC graph for n ≡ 0, 1, 3(mod 4).

Proof. Let V (Cn) = {vj | 1 ≤ j ≤ n}. It is to be noted that, |V (Cn)| = n = |E(Cn)|.Consider a bijection f : V (Cn)→ {1, 2, 3, . . . , n} defined as below.

f(vj) =

j; j ≡ 0, 1(mod 4).

j + 1; j ≡ 2(mod 4).

j − 1; j ≡ 3(mod 4), 1 ≤ j ≤ n.



n ≡ 0(mod 4) ef (0) = ef (1) = n2

n ≡ 1(mod 4) ef (0) =⌊n2

⌋, ef (1) =

⌈n2

⌉

n ≡ 3(mod 4) ef (0) =⌈n2

⌉, ef (1) =

⌊n2

⌋

91


Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Cn is SDC graph.

Example 5.3.1. SDC labeling in cycle C5 is demonstrated in the following Figure

5.2.

1

3

2 4

5

v1

v2

v3v4

v5

Figure 5.2: SDC labeling in C5

Theorem 5.3.2. Cn with one chord is SDC graph.

Proof. Let cycle Cn with one chord be denoted as G.

Let V (G) = {vj | 1 ≤ j ≤ n} and E(G) = {vjvj+1 | 1 ≤ j ≤ n−1}⋃{vnv1}⋃{v2vn},

where v2vn is a chord.

Also it is to be noted that, |V (G)| = n and |E(G)| = n+ 1.

Consider a bijection f : V (G)→ {1, 2, 3, . . . , n} defined as below.

Case 1: n ≡ 0, 1, 2(mod 4).

f(vj) =

j; j ≡ 1, 2(mod 4).

j + 1; j ≡ 3(mod 4).

j − 1; j ≡ 0(mod 4), 1 ≤ j ≤ n.

Case 2: n ≡ 3(mod 4).

f(vj) =

j; j ≡ 1, 2(mod 4).

j + 1; j ≡ 3(mod 4).

j − 1; j ≡ 0(mod 4), 1 ≤ j ≤ n− 2.

f(vn−1) = n.

f(vn) = n− 1.


92



n ≡ 0, 2, 3(mod 4) ef (1) =⌊n+12

⌋, ef (0) =

⌈n+12

⌉

n ≡ 1(mod 4) ef (0) = ef (1) = n+12

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Cn with one chord is SDC graph.

Example 5.3.2. SDC labeling in C6 with one chord is demonstrated in the following

Figure 5.3.

v1

1

2

4

3

5

6

v3

v4

v5

v6v2

Figure 5.3: SDC labeling in C6 with one chord

Theorem 5.3.3. Cycle with twin chords Cn,3 is SDC graph.

Proof. Let V (Cn,3) = {vj | 1 ≤ j ≤ n}.Let E(Cn,3) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}

⋃{v2vn}⋃{v2vn−1}, where vnv2,

v2vn−1 are the chords.

It is to be noted that, |V (Cn,3)| = n and |E(Cn,3)| = n+ 2.

Consider a bijection f : V (Cn,3)→ {1, 2, 3, . . . , n} defined as below.

Case 1: n ≡ 0(mod 4).

f(vj) =

j; j ≡ 1, 0(mod 4).

j + 1; j ≡ 2(mod 4).

j − 1; j ≡ 0(mod 4) 1 ≤ j ≤ n.

Case 2: n ≡ 1, 2(mod 4).

f(vi) =

j; j ≡ 1, 2(mod 4).

j + 1; j ≡ 3(mod 4).

j − 1; j ≡ 0(mod 4), 1 ≤ j ≤ n.

93


Case 3: n ≡ 3(mod 4).

f(vj) =

j; j ≡ 1, 2(mod 4).

j + 1; j ≡ 3(mod 4).

j − 1; j ≡ 0(mod 4), 1 ≤ j ≤ n− 2.

f(vn−1) = n.

f(vn) = n− 1.



n ≡ 0, 2(mod 4) ef (1) = n+22

= ef (0)

n ≡ 1(mod 4) ef (0) =⌊n+22

⌋, ef (1) =

⌈n+22

⌉

n ≡ 3(mod 4) ef (1) =⌊n+22

⌋, ef (0) =

⌈n+22

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Cn,3 is SDC graph.

Example 5.3.3. SDC labeling in C7,3 is demonstrated in the following Figure 5.4.

1

2

4

3 5

6

7

v7

v6

v5v4

v3

v2

v1

Figure 5.4: SDC labeling in C7,3

Theorem 5.3.4. Cn(1, 1, n− 5) is SDC graph for n ≡ 0, 1, 2(mod 4).

Proof. Let V (Cn(1, 1, n− 5)) = {vj | 1 ≤ j ≤ n}.Let E(Cn(1, 1, n−5)) = {vjvj+1 | 1 ≤ j ≤ n−1}⋃{vnv1}

⋃{v1v3}⋃{v3vn−1}

⋃{vn−1v1},where v1vn−1, v1v3, vn−1v3 are the chords of Cn(1, 1, n− 5).

It is to be noted that, |V (Cn(1, 1, n− 5))| = n and |E(Cn)| = n+ 3.

Consider a bijection f : V (Cn(1, 1, n− 5))→ {1, 2, 3, . . . , n} defined as below.

94


Case 1: n ≡ 0, 1(mod 4).

f(vj) =

j; j ≡ 1, 2(mod 4).

j + 1; j ≡ 3(mod 4).

j − 1; j ≡ 0(mod 4), 1 ≤ j ≤ n.

Case 2: n ≡ 2(mod 4).

f(vj) =

j; j ≡ 1, 0(mod 4).

j + 1; j ≡ 2(mod 4).

j − 1; j ≡ 3(mod 4), 1 ≤ j ≤ n− 2.

f(vn−1) = n.

f(vn) = n− 1.



n ≡ 1(mod 4) ef (0) = n+32

= ef (1)

n ≡ 0(mod 4) ef (1) =⌊n+32

⌋, ef (0) =

⌈n+32

⌉

n ≡ 2(mod 4) ef (0) =⌊n+32

⌋, ef (1) =

⌈n+32

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Cn(1, 1, n− 5) is SDC graph.

Example 5.3.4. SDC labeling in C8(1, 1, 3) is demonstrated in the following Figure

5.5.

1

2

4

3

5

6

8

7

v3

v4

v5

v6

v7

v8v2

v1

Figure 5.5: SDC labeling in C8(1, 1, 3)

95


Theorem 5.3.5. Wn is SDC graph for n ≡ 0, 1, 2(mod 4).

Proof. Let V (Wn) = {v0, vj, | 1 ≤ j ≤ n}, where vj(1 ≤ j ≤ n) are rim vertices and

v0 is the apex vertex.


It is to be noted that, |V (Wn)| = n+ 1 and |E(Wn)| = 2n.

Consider a bijection f : V (Wn)→ {1, 2, 3, . . . , n+ 1} defined as below.

Case 1: n ≡ 1(mod 4).

f(v0) = 1.

f(vj) =

j; j ≡ 3(mod 4).

j + 1; j ≡ 0, 1(mod 4).

j + 2; j ≡ 2(mod 4), 1 ≤ j ≤ n.

Case 2: n ≡ 0(mod 4).

f(v0) = 1.

f(vj) =

j; j ≡ 3(mod 4).

j + 1; j ≡ 0, 1(mod 4).

j + 2; j ≡ 2(mod 4), 1 ≤ j ≤ n.

Case 3: n ≡ 2(mod 4).

f(v0) = 2.

f(vj) =

j; j ≡ 1(mod 4).

j + 1; j ≡ 2, 3(mod 4).

j + 2; j ≡ 0(mod 4), 1 ≤ j ≤ n.



n ≡ 0, 1, 2(mod 4) ef (0) = n = ef (1)

Then we get, |ef (0)− ef (1)| ≤ 1.

96


That is, Wn is SDC graph.

Example 5.3.5. SDC labeling in W5 is demonstrated in the following Figure 5.6.

v1

v2

v3 v4

v5

1

2

4

3 5

6v0

Figure 5.6: SDC labeling in W5

Theorem 5.3.6. The helm Hn is SDC.

Proof. Let V (Hn) = {v0, vj, uj | 1 ≤ j ≤ n}, where v0 is the apex vertex, d(vj) =

4(1 ≤ j ≤ n) and d(uj) = 1(1 ≤ j ≤ n).

Let E(Hn) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{v0vj | 1 ≤ j ≤ n}⋃{vjuj | 1 ≤

j ≤ n}.It is to be noted that, |V (Hn)| = 2n+ 1 and |E(Hn)| = 3n.

Consider a bijection f : V (Hn)→ {1, 2, 3, . . . , 2n+ 1} defined as below.

f(v0) = 1.

To label the vertices vj, uj(1 ≤ j ≤ n), let us consider the below possibilities.

Case 1: n ≡ 0, 2(mod 4).

f(v2j−1) = 4j − 1; 1 ≤ j ≤ n

2.

f(v2j) = 4j − 2; 1 ≤ j ≤ n

2.

f(uj) = f(vj) + 2; 1 ≤ j ≤ n.

Case 2: n ≡ 1, 3(mod 4).

f(v2j−1) = 4j − 1; 1 ≤ j ≤ n− 1

2.

f(v2j) = 4j − 2; 1 ≤ j ≤ n− 1

2.

97


f(uj) = f(vj) + 2; 1 ≤ j ≤ n− 1.

f(vn) = 2n+ 1.

f(un) = 2n.



n ≡ 1, 3(mod 4) ef (0) =⌊3n2

⌋, ef (1) =

⌈3n2

⌉

n ≡ 0, 2(mod 4) ef (0) = 3n2

= ef (1)

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Hn is SDC graph.

Example 5.3.6. SDC labeling in H6 is demonstrated in the following Figure 5.7.

1

3

5

7

9

2

4

6

8

11

13

12

14

v3

v4

v5

v1

v2

u3

u4

u5

u6

u1

u2

v0

v6

Figure 5.7: SDC labeling in H6

Theorem 5.3.7. The web graph Wbn is SDC graph.

Proof. Let V (Wbn) = {u0, uj, vj, wj | 1 ≤ j ≤ n}, where u0 is the apex vertex,

uj(1 ≤ j ≤ n) are the vertices corresponding to inner cycle, vj(1 ≤ j ≤ n) are the

vertices corresponding to outer cycle and d(wj) = 1(1 ≤ j ≤ n) are vertices of Wbn.

Let E(Wbn) = {ujuj+1, vjvj+1 | 1 ≤ j ≤ n − 1}⋃{unu1, vnv1}⋃{u0uj, ujvj, vjwj |

1 ≤ j ≤ n}.It is to be noted that, |V (Wbn)| = 3n+ 1 and |E(Wbn)| = 5n.

98


Consider a bijection f : V (Wbn)→ {1, 2, 3, . . . , 3n+ 1} defined as below.

f(u0) = 1.

f(uj) = 2j; 1 ≤ j ≤ n.

f(vj) = 2j + 1; 1 ≤ j ≤ n.

f(wj) = (2n+ 1) + j; 1 ≤ j ≤ n.



n ≡ 0, 2(mod 4) ef (0) = 5n2

= ef (1)

n ≡ 1, 3(mod 4) ef (1) =⌊5n2

⌋, ef (0) =

⌈5n2

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Wbn is SDC graph.

Example 5.3.7. SDC labeling in Wb5 is demonstrated in the following Figure 5.8.

v1

v2

v3 v4

v5

1

2

4

3

5

6 8

10

79

11

12

13

1415

16

u0

u1

u2

u3

u5

u4

w1

w2

w3 w4

w5

Figure 5.8: SDC labeling in Wb5

Theorem 5.3.8. Shell Sn is SDC graph.

Proof. Let V (Sn) = {vj | 1 ≤ j ≤ n}, where v1 is apex vertex and and vj(2 ≤ j ≤ n)

99


are other vertices of shell graph Sn such that

deg(vj) =

2; if j = 2, n.

3; if 3 ≤ j ≤ n− 1.

Let E(Sn) = {vjvj+1 | 1 ≤ j ≤ n− 1}⋃{vnv1}⋃{v1vj | 2 ≤ j ≤ n− 1}.

It is to be noted that, |V (Sn)| = n and |E(Sn)| = 2n− 3.

Consider a bijection f : V (Sn)→ {1, 2, 3, . . . , n} defined as below.

Case 1: n ≡ 1, 3(mod 4)

f(vj) =

j; j ≡ 1, 2(mod 4).

j + 1; j ≡ 3(mod 4).

j − 1; j ≡ 0(mod 4), 1 ≤ j ≤ n− 1.

f(vn) = n.

Case 2: n ≡ 0, 2(mod 4)

f(vj) =

j; j ≡ 1, 2(mod 4).

j + 1; j ≡ 3(mod 4).

j − 1; j ≡ 0(mod 4), 1 ≤ j ≤ n.



n ≡ 0, 2, 3(mod 4) ef (1) =⌊2n−3

2

⌋, ef (0) =

⌈2n−3

2

⌉

n ≡ 1(mod 4) ef (1) =⌈2n−3

2

⌉, ef (0) =

⌊2n−3

2

⌋

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Sn is SDC graph.

Example 5.3.8. SDC labeling in S7 is demonstrated in the following Figure 5.9.

100


v3

v4 v5

v6

v1

v7

1

2

3

4

5

6

7v2

Figure 5.9: SDC labeling in S7

Theorem 5.3.9. Fln is SDC graph.

Proof. Let V (Fln) = {v0, vj, uj | 1 ≤ j ≤ n}, where v0 is the apex vertex, d(vj) =

4(1 ≤ i ≤ n) and d(uj) = 2(1 ≤ i ≤ n).

Let E(Fln) = {vjvj+1 | 1 ≤ j ≤ n− 1}⋃{vnv1}⋃{v0vj, vjuj, v0uj | 1 ≤ j ≤ n}.

It is to be noted that, |V (Fln)| = 2n+ 1 and |E(Fln)| = 4n.

Consider a bijection f : V (Fln)→ {1, 2, 3, . . . , 2n+ 1} defined as below.

f(v0) = 1.

f(vj) = 2j; 1 ≤ j ≤ n.

f(uj) = 2j + 1; 1 ≤ j ≤ n.


ef (0) = 2n = ef (1)

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Fln is SDC graph.

Example 5.3.9. SDC labeling in Fl4 is demonstrated in the following Figure 5.10.

101


u1

u2

u3

u4

v1

v2

v3

v4

1

3

2

5 4

7

6

98v0

Figure 5.10: SDC labeling in Fl4

Theorem 5.3.10. Double fan DFn is SDC graph.

Proof. Let V (DFn) = {u,w, vj | 1 ≤ j ≤ n}, where d(u) = n = d(w) are apex

vertices and uj(1 ≤ j ≤ n) are the vertices on the path Pn corresponding to DFn.

Let E(DFn) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{uuj, wuj | 1 ≤ j ≤ n}.It is to be noted that, |V (DFn)| = n+ 2 and |E(DFn)| = 3n− 1.

Consider a bijection f : V (DFn)→ {1, 2, 3, . . . , n+ 2} defined as below.

f(u) = 1.

f(w) = 2.

Case 1: n ≡ 0, 1, 3(mod 4).

f(uj) =

j + 1; j ≡ 3(mod 4)

j + 2; j ≡ 1, 0(mod 4)

j + 3; j ≡ 2(mod 4), 1 ≤ j ≤ n.

Case 2: n ≡ 2(mod 4).

f(uj) =

j + 1; j ≡ 3(mod 4)

j + 2; j ≡ 1, 0(mod 4)

j + 3; j ≡ 2(mod 4), 1 ≤ j ≤ n− 1.

f(un) = n+ 2.

102

5.4. SDC Labeling of Snakes Related Graphs



n ≡ 1, 3(mod 4) ef (0) = 3n−12

= ef (1)

n ≡ 0, 2(mod 4) ef (1) =⌊3n−1

2

⌋, ef (0) =

⌈3n−1

2

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, DFn is SDC graph.

Example 5.3.10. SDC labeling in DF5 is demonstrated in the following Figure

5.11.

u1 u2 u3 u4 u5

u

w

1

3

2

4 65 7

Figure 5.11: SDC labeling in DF5

5.4 SDC Labeling of Snakes Related Graphs

Vaidya and Shah[52] illuminated some fascinating snakes related graphs for cordial

labeling. Getting inspired by this work, in this section we have emanated some

snakes related graphs for SDC labeling.

Theorem 5.4.1. Triangular snake Tn is SDC graph for n ≡ 0, 1, 2 (mod 4).

Proof. Let Pn be a path with V (Pn) = {vj | 1 ≤ j ≤ n} and E(Pn) = {vjvj+1 | 1 ≤j ≤ n− 1}.To construct triangular snake graph Tn from the path Pn, join vj and vj+1 to new

vertex uj by edges e′2j−1 = vjuj and e′2j = vj+1uj, i = 1, 2, . . . , n− 1.

It is to be noted that, |V (Tn)| = 2n− 1 and |E(Tn)| = 3n− 3.

Let us a bijection f : V (Tn)→ {1, 2, 3, . . . , 2n− 1} defined as below.

103


Case 1: n ≡ 0, 1(mod 4).

f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4) 1 ≤ j ≤ n.

f(ui) = n+ i; 1 ≤ i ≤ n− 1.

Case 2: n ≡ 2(mod 4).

f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4) 1 ≤ j ≤ n.

f(u1) = n.

f(uj) = n+ j; 2 ≤ j ≤ n− 1.



n ≡ 0(mod 4) ef (1) =⌈3n−3

2

⌉, ef (0) =

⌊3n−3

2

⌋

n ≡ 1(mod 4) ef (1) = 3n−32

= ef (0)

n ≡ 2(mod 4) ef (1) =⌊3n−3

2

⌋, ef (0) =

⌈3n−3

2

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Tn is SDC graph.

Example 5.4.1. SDC labeling in T6 is demonstrated in the following Figure 5.12.

v1 v2 v3 v4 v5 v6

1

u1 u2 u3 u4 u5

2 4

6 108 11

3 5 7

9

Figure 5.12: SDC labeling in T6

Theorem 5.4.2. Double triangular snake DTn is SDC graph.

104


Proof. Let Pn be a path with V (Pn) = {vj | 1 ≤ j ≤ n} and E(Pn) = {vjvj+1 | 1 ≤j ≤ n− 1}.To construct DTn, join vj and vj+1 to the new vertices uj, wj by edges e′2j−1 =

ujvj, e′2j = ujvj+1, e

′′2j−1 = wjvj, e

′′2j = wjvj+1, j = 1, 2, . . . , n− 1.

It is to be noted that, |V (DTn)| = 3n− 2 and |E(DTn)| = 5n− 5.

Consider a bijection f : V (DTn)→ {1, 2, 3, . . . , 3n− 2} defined as below.

f(vj) =

j; j ≡ 1, 0(mod 4)

j + 1; j ≡ 2(mod 4)

j − 1; j ≡ 3(mod 4) 1 ≤ j ≤ n.

To label the vertices {uj, wj | 1 ≤ j ≤ n− 1}, let us consider the below possibilities.

Case 1: n ≡ 0, 1, 2(mod 4).

f(uj) = n+ j; 1 ≤ j ≤ n− 1.

f(wj) = (2n− 1) + i; 1 ≤ j ≤ n− 1.

Case 2: n ≡ 3(mod 4).

f(u1) = n+ 2.

f(u2) = n+ 1.

f(uj) = n+ j; 3 ≤ j ≤ n− 1.

f(wj) = (2n− 1) + j 1 ≤ j ≤ n− 1.



n ≡ 1, 3(mod 4) ef (1) = 5n−52

= ef (0)

n ≡ 0(mod 4) ef (1) =⌊5n−5

2

⌋, ef (0) =

⌈5n−5

2

⌉

n ≡ 2(mod 4) ef (1) =⌈5n−5

2

⌉, ef (0) =

⌊5n−5

2

⌋

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, DTn is SDC graph.

Example 5.4.2. SDC labeling in DT5 is demonstrated in the following Figure 5.13.

105


3

w1 w2 w3 w4

v1v2 v3 v4 v5

1 2 54

u1 u2 u3 u4

6 7 98

10 11 12 13

Figure 5.13: SDC labeling in DT5

Theorem 5.4.3. Quadrilateral snake Qn is SDC graph.

Proof. Let Pn be a path with V (Pn) = {vj | 1 ≤ j ≤ n} and E(Pn) = {vjvj+1 | 1 ≤j ≤ n− 1}.To construct Qn, join vj and vj+1 to the new vertices uj, wj by edges e′2j−1 =

vjuj, e′2j = vj+1wj and e′′j = ujwj, j = 1, 2, . . . n− 1.

It is to be noted that, |V (Qn)| = 3n− 2 and |E(Qn)| = 4n− 4.

Consider a bijection f : V (Qn)→ {1, 2, 3, . . . , 3n− 2} defined as below.

f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4) 1 ≤ j ≤ n.

To label the vertices {uj, wj | 1 ≤ j ≤ n− 1}, let us consider the below possibilities.

Case 1: n ≡ 0(mod 4).

f(uj) = n+ 2j − 1; 1 ≤ j ≤ n− 1.

f(wj) = n+ 2j; 1 ≤ j ≤ n− 1.

Case 2: n ≡ 2(mod 4).

f(u1) = n,

f(uj) = n+ 2j − 1; 2 ≤ j ≤ n− 1.

f(wj) = n+ 2j; 1 ≤ j ≤ n− 1.

106


In above two cases, whenever j ≡ 0(mod 4) interchange f(uj) and f(wj).

Case 3: n ≡ 1(mod 4).

f(uj) = n+ 2j − 1; 1 ≤ j ≤ n− 1.

f(wj) = n+ 2j; 1 ≤ j ≤ n− 1.

Case 4: n ≡ 3(mod 4).

f(uj) = n+ 2j − 1; 1 ≤ j ≤ n− 1.

f(wj) = n+ 2j; 1 ≤ j ≤ n− 1.

In above two cases, whenever j ≡ 2(mod 4) interchange f(uj) and f(wj)


ef (1) = 2n− 2 = ef (0)

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Qn is SDC graph.

Example 5.4.3. SDC labeling in Q5 is demonstrated in the following Figure 5.14.

11

2 4

6 108 12

1 3 5

7 9

v1 v2 v3 v4 v5

w1 w2u1 u2 w3u3 w4u4

13

Figure 5.14: SDC labeling in Q5

Theorem 5.4.4. Double quadrilateral snake DQn is SDC graph.

Proof. Let Pn be a path with V (Pn) = {vj | 1 ≤ j ≤ n} and E(Pn) = {vjvj+1 | 1 ≤j ≤ n− 1}.To construct DQn, join vj and vj+1 to new vertices uj, u

′j, wj and w′j by edges

e(u)2j−1 = viuj, e

(u)2j = vj+1u

′j, e

(uu)j = uju

′j, e

(w)2j−1 = vjwj, e

(w)2j = vj+1w

′j, e

(ww)j = wjw

′j

for j = 1, 2, . . . , n− 1.

107


It is to be noted that, |V (DQn)| = 5n− 4 and |E(DQn)| = 7n− 7.

Consider a bijection f : V (DQn)→ {1, 2, 3, . . . , 5n− 4} defined as below.

f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4) 1 ≤ j ≤ n.

To label the vertices {uj, u′j | 1 ≤ j ≤ n − 1} and {wj, w′j | 1 ≤ j ≤ n − 1}, let us

consider the below possibilities.

Case 1: n ≡ 0, 1, 3(mod 4).

f(uj) = n+ 2j − 1; 1 ≤ j ≤ n− 1.

f(wj) = n+ 2j; 1 ≤ j ≤ n− 1.

f(u′j) = 3n− 3 + 2j; 1 ≤ j ≤ n− 1.

f(w′j) = 3n− 2 + 2j; 1 ≤ j ≤ n− 1.

Case 2: n ≡ 2(mod 4).

f(u1) = n, f(u′1) = n+ 2,

f(uj) = n+ 2j − 1; 2 ≤ j ≤ n− 1.

f(wj) = n+ 2j; 2 ≤ j ≤ n− 1.

f(u′j) = 3n− 3 + 2j; 1 ≤ j ≤ n− 1.

f(w′j) = 3n− 2 + 2j; 1 ≤ j ≤ n− 1.

For n ≡ 0, 2(mod 4) : whenever j ≡ 0(mod 4) interchange f(uj) with f(wj) and

f(u′j) with f(w′j).

For n ≡ 1, 3(mod 4) : whenever j ≡ 2(mod 4) interchange f(uj) with f(wj) and

f(u′j) with f(w′j).



n ≡ 1, 3(mod 4) ef (1) = 7(n−1)2

= ef (0)

n ≡ 0, 2(mod 4) ef (1) =⌊7(n−1)

2

⌋, ef (0) =

⌈7(n−1)

2

⌉

108


Then we get, |ef (0)− ef (1)| ≤ 1.

That is, DQn is SDC graph.

Example 5.4.4. SDC labeling in DQ5 is demonstrated in the following Figure 5.15.

v1 v2 v3 v4 v5

w1 w2

2

u1 u2 w3u3 w4u4

2 41 3 5

116 108 127 9 13

14 15 17 16 18 19 20 21

w'1 w'2u'1 u'2 w'3u'3 w'4u'4

Figure 5.15: SDC labeling in DQ5

Theorem 5.4.5. Alternate triangular snake A(Tn) confesss SDC labeling.

Proof. Let Pn be a path with V (Pn) = {vj | 1 ≤ j ≤ n} and E(Pn) = {vjvj+1 | 1 ≤j ≤ n− 1}.To construct A(Tn), join v2i−1 and v2i to new vertex ui, where 1 ≤ i ≤ bn

2c.

Therefore V (A(Tn)) = {vi, uj/1 ≤ i ≤ n, 1 ≤ j ≤ bn2c}.

It is to be noted that

|V (A(Tn))| =

3n2

; if n is even.

3n−12

; if n is odd.

and

|E(A(Tn))| =

2n− 1 ; if n is even.

2n− 2 ; if n is odd.

Consider a bijection f : V (A(Tn))→ {1, 2, 3, . . . , |V (A(Tn))|} defined as below.

f(vi) =

i+ 2 ; i ≡ 1(mod 4)

i ; i ≡ 2(mod 4)

i+ 1 ; i ≡ 0, 3(mod 4) 1 ≤ i ≤ n.

f(u1) = 1.

109


Case 1: n ≡ 0, 2, 3(mod 4).

f(ui) = n+ i; 2 ≤ i ≤⌊n

2

⌋.

Case 2: n ≡ 1(mod 4).

f(u2) = n+ 1,

f(ui) = n+ i; 3 ≤ i ≤ n− 1

2.



n ≡ 1, 3(mod 4) ef (1) = n− 1 = ef (0)

n ≡ 0, 2(mod 4) ef (1) =⌊2n−1

2

⌋, ef (0) =

⌈2n−1

2

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, A(Tn) is SDC graph.

Example 5.4.5. SDC labeling in A(T7) is demonstrated in the following Figure

5.16.

u1 u2 u3

v1 v2 v3 v4 v5 v6 v7

3

1

2 6754 8

9 10

Figure 5.16: SDC labeling in A(T7)

Theorem 5.4.6. Alternate quadrilateral snake A(Qn) confesss SDC labeling.

Proof. Let Pn be a path with V (Pn) = {vj | 1 ≤ j ≤ n} and E(Pn) = {vjvj+1 | 1 ≤j ≤ n− 1}.Then A(Qn) will be constructed by joining v2i−1 and v2i to new vertices ui and wi

respectively and then joining ui and wi, where 1 ≤ i ≤ bn2c.

Therefore V (A(Qn)) = {vi, uj, wj/1 ≤ i ≤ n, 1 ≤ j ≤ bn2c}.

110



|V (A(Qn))| =

2n; if n is even.

2n− 1; if n is odd.

and

|E(A(Qn))| =

5n−22

; if n is even.

5n−52

; if n is odd.

Consider a bijection f : V (A(Qn))→ {1, 2, 3, . . . , |V (A(Qn))|} defined as below.

Case 1: n ≡ 0, 2(mod 4).

f(vi) =

2i− 1; i ≡ 1, 2, 3, 4(mod 8)

2i; i ≡ 0, 5, 6, 7(mod 8) 1 ≤ i ≤ n.

f(ui) =

4i− 2; i ≡ 1, 2, (mod 4)

4i− 3; i ≡ 3, 4(mod 4) 1 ≤ i ≤ n2.

f(wi) =

4i; ; i ≡ 1, 2, (mod 4)

4i− 1; i ≡ 3, 4(mod 4) 1 ≤ i ≤ n2.

Case 2: n ≡ 1, 3(mod 4).

f(vi) =

2i− 1 ; i ≡ 1, 2, 3, 4(mod 8)

2i ; i ≡ 0, 5, 6, 7(mod 8) 1 ≤ i ≤ n− 1.

f(vn) = 2n− 1.

f(ui) =

4i− 2; i ≡ 1, 2, (mod 4)

4i− 3; i ≡ 3, 4(mod 4) 1 ≤ i ≤ n−12.

f(wi) =

4i; i ≡ 1, 2, (mod 4)

4i− 1; i ≡ 3, 4(mod 4) 1 ≤ i ≤ n−12.


111



n ≡ 1(mod 4) ef (1) = 5n−52

= ef (0)

n ≡ 3(mod 4) ef (1) =⌊5n−5

2

⌋, ef (0) =

⌈5n−5

2

⌉

n ≡ 2(mod 4) ef (1) = 5n−22

= ef (0)

n ≡ 0(mod 4) ef (1) =⌈5n−2

2

⌉, ef (0) =

⌊5n−2

2

⌋

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, A(Qn) is SDC graph.

Example 5.4.6. SDC labeling in A(Q8) is demonstrated in the following Figure

5.17.

v1 v2 v3 v4 v5 v6 v7 v8

w1 w2 w3 w4u1 u2 u3 u4

1

2 4

3 5 7

6

10

8 119

12

13

14

15

16

Figure 5.17: SDC labeling in A(Q8)

Theorem 5.4.7. Double alternate triangular snake DA(Tn) confesss SDC labeling.

Proof. Let Pn be a path with V (Pn) = {vj | 1 ≤ j ≤ n} and E(Pn) = {vjvj+1 | 1 ≤j ≤ n− 1}.To construct DA(Tn), join v2i−1 and v2i to new vertices ui and wi respectively,

1 ≤ i ≤ bn2c.

Then V (DA(Tn))) = {vi, uj, wj | 1 ≤ i ≤ n, 1 ≤ j ≤ bn2c}.


|V (DA(Tn))| =

2n; if n is even.


and

|E(DA(Tn))| =

3n− 1; if n is even.


112


Consider a bijection f : V (DA(Tn))→ {1, 2, 3, . . . , |V (DA(Tn))|} defined as below.

f(vi) =

i+ 2; i ≡ 1(mod 4)

i; i ≡ 2(mod 4)

i+ 1; i ≡ 0, 3(mod 4) 1 ≤ i ≤ n.

f(u1) = 1.

To label the vertices {ui | 2 ≤ i ≤ n − 1} and {wi | 1 ≤ i ≤ n − 1}, consider the

labeling defined by means of the below cases.

Case 1: n ≡ 0, 2(mod 4).

f(ui) = n+ i; 2 ≤ i ≤ n

2.

f(wi) =3n

2+ i; 1 ≤ i ≤ n

2.

Case 2: n ≡ 1(mod 4).

f(u2) = n+ 1.

f(ui) = n+ i; 3 ≤ i ≤ n− 1

2,

f(wi) =3(n− 1)

2+ 1 + i; 1 ≤ i ≤ n− 1

2.

Case 3: n ≡ 3(mod 4).

f(ui) = n+ i; 2 ≤ i ≤ n− 1

2,

f(wi) =3(n− 1)

2+ 1 + i; 1 ≤ i ≤ n− 1

2.



n ≡ 1, 3(mod 4) ef (1) = 3n−32

= ef (0)

n ≡ 0, 2(mod 4) ef (1) =⌊3n−1

2

⌋, ef (0) =

⌈3n−1

2

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, DA(Tn) is SDC graph.

113


Example 5.4.7. SDC labeling in DA(T10) is demonstrated in the following Figure

5.18.

u1 u2 u3 u4 u5

1

2 4v1 v2 v3 v4 v5 v6 v7 v8 v9 v10

3 675 98 1011

13 14 1512

17 201816 19

w1 w2 w3 w4 w5

Figure 5.18: SDC labeling in DA(T10)

Theorem 5.4.8. Double alternate quadrilateral snake DA(Qn) confesss SDC label-

ing.

Proof. Let Pn be a path with V (Pn) = {vj | 1 ≤ j ≤ n} and E(Pn) = {vjvj+1 | 1 ≤j ≤ n− 1}.To construct DA(Qn), join v2i−1 and v2i to new vertices ui, wi and u′i, w

′i respectively

and adding the edges uiwi and u′iw′i, 1 ≤ i ≤ bn

2c.

Then V (DA(Qn)) = {vi, uj, wj, u′j, w′j | 1 ≤ i ≤ n, 1 ≤ j ≤ bn2c}.


|V (DA(Qn))| =

3n; if n is even.


and

|E(DA(Qn))| =

4n− 1; if n is even.


Consider a bijection f : V (DA(Qn))→ {1, 2, 3, . . . , |V (DA(Qn))|} defined as below.

Case 1: n ≡ 0, 2(mod 4).

f(vi) =

2i− 1; i ≡ 1, 2, 3, 4(mod 8)

2i; i ≡ 0, 5, 6, 7(mod 8) 1 ≤ i ≤ n.

114


f(u′1) = 2n+ 1.

f(u′2) = 2n+ 2.

f(u′i) = 2n+ 2i− 1; 3 ≤ i ≤⌊n

2

⌋.

f(w′1) = 2n+ 3,

f(w′i) = 2n+ 2i; 2 ≤ i ≤⌊n

2

⌋.

Case 2: n ≡ 1, 3(mod 4).

f(vi) =

2i− 1; i ≡ 1, 2, 3, 4(mod 8)

2i; i ≡ 0, 5, 6, 7(mod 8) 1 ≤ i ≤ n− 1.

f(vn) = 3n− 2.

f(u′1) = 2n− 1.

f(u′2) = 2n.

f(u′i) = 2(n− 1) + 2i− 1; 3 ≤ i ≤⌊n

2

⌋.

f(w′1) = 2n+ 1.

f(w′i) = 2(n− 1) + 2i; 2 ≤ i ≤⌊n

2

⌋.

For n ≡ 0, 1, 2, 3(mod 4).

f(ui) =

4i− 2; i ≡ 1, 2, (mod 4)

4i− 3; i ≡ 3, 4(mod 4) 1 ≤ i ≤⌊n2

⌋.

f(wi) =

4i; i ≡ 1, 2, (mod 4)

4i− 1; i ≡ 3, 4(mod 4) 1 ≤ i ≤⌊n2

⌋.



n ≡ 1, 3(mod 4) ef (1) = 2n− 2 = ef (0)

n ≡ 0, 2(mod 4) ef (1) =⌊4n−1

2

⌋, ef (0) =

⌈4n−1

2

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, DA(Qn) is SDC graph.

115


Example 5.4.8. SDC labeling in DA(Q9) is demonstrated in the following Figure

5.19.

w1 w2 w3 w4u1 u2 u3 u4

v1v2 v3 v4 v5 v6 v7 v8

w'1 w'2 w'3 w'4u'1 u'2 u'3 u'4

1

2 4

3 5 7

6

10

8 119

12

13

14

15

16

17 1819 20 21 22 23 24

25v9

Figure 5.19: SDC labeling in DA(Q9)


In this chapter we have emanated some new standard SDC graphs and snake related

SDC graphs.

However, there is no such standard relation between either of the two DC and SDC

labelings; like, we may find a graph which confess one labeling but not the other.

To observe this matter more effectively, the list of graph families satisfying/ not

satisfying certain labeling is shown below.

• The star K1,n is both DC and SDC (Refer [44] and [1]).

• The triangular snake graph DT5 is not DC (easy to check) but it is SDC (Refer

Theorem 5.4.2).

• The triangular snake graph T7 is neither DC (easy to check) nor SDC (Refer

Theorem 5.4.1).

At the end, we give some problems.


be SDC graph.

Problem 5.5.2. To construct similar results for other graph families and to make

the relation stonger between the two labeling mentioned.

116


The penultimate chapter is also based on SDC graphs where SDC labeling is going

to be discussed for the graphs obtained by using graph operations.

117

CHAPTER 6

Sum Divisor Cordial Labeling

With the Use of Some Graph

Operations

SDC labeling have been discussed for certain graph families in the earlier Chapter-5,

while the existing chapter aims to give a brief account of SDC labeling in the graphs

constructed by using the following graph operations.

z Ringsum of different graphs with star graph K1,n.

z Corona of different graphs with graph K1.

z Vertex switching of graphs.

z Duplication a vertex in star, cycle and path allied graphs.

6.1 SDC Labeling of Graphs With the Use of Ringsum of

Different Graphs with Star Graph K1,n

Ghodasara and Rokad[5] illuminated and derived some fascinating results on cordial


Under the inspiration of this credibility, in the current segment we demonstrate

some results on SDC labeling of the graphs constructed from the graph operation

ringsum.

118

6.1. SDC Labeling of Graphs With the Use of Ringsum of Different Graphs with Star Graph K1,n

Remark 6.1.1. Throughout this chapter we consider the ringsum of a graph G with

K1,n by considering any one vertex of G and apex vertex of K1,n as a common vertex.

Theorem 6.1.1. Cn ⊕K1,n is SDC.

Proof. Let V (Cn ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (Cn) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.

Let E(Cn ⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤ n}.

It is to be noted that, |V (Cn ⊕K1,n)| = |E(Cn ⊕K1,n)| = 2n.

Consider a bijection f from V (Cn ⊕K1,n) to {1, 2, 3, . . . , 2n} defined as below.

f(uj) = 2j − 1; 1 ≤ j ≤ n.

f(vk) = 2k; 1 ≤ k ≤ n.

As per this pattern, allocate the vertex labels such that for any edge uiui+1 ∈E(Cn ⊕K1,n),

f(ui) | f(ui+1), 1 ≤ i ≤ n− 1

and

f(u1) - f(vj) 1 ≤ j ≤ n.


ef (1) = ef (0) = n.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Cn ⊕K1,n is SDC graph.

Example 6.1.1. SDC labeling in the graph C5⊕K1,5 is demonstrated in the following

Figure 6.1.

119


1

v3 v5v1 v2 v4

3

2 4

5

6

7

8

9

10

u1

u2

u3

u5

u4

Figure 6.1: SDC labeling in C5 ⊕K1,5

Theorem 6.1.2. G⊕K1,n is sum DC graph, where G is cycle Cn with one chord.


Let V (G ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (G) = V (Cn) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.


where u2un is the chord of Cn and vertices u1, u2, un form a triangle with chord u2un.

Also it is to be noted that, |V (G⊕K1,n)| = 2n and |E(G⊕K1,n)| = 2n+ 1.

Consider a bijection f from V (G⊕K1,n) to {1, 2, 3, . . . , 2n} defined as below.

f(uj) = 2j − 1; 1 ≤ j ≤ n.

f(vk) = 2k; 1 ≤ k ≤ n.

As per this pattern, allocate the vertex labels such that for any edge ujuj+1 ∈E(G⊕K1,n),

f(uj) | f(uj+1) 1 ≤ j ≤ n− 1

and f(u1) - f(vk) 1 ≤ k ≤ n.


ef (0) = n, ef (1) = n+ 1.

Then we get, |ef (0)− ef (1)| ≤ 1.

120


That is, G⊕K1,n is SDC graph, where G is cycle Cn with one chord.

Example 6.1.2. SDC labeling in the graph constructed from ringsum of C7 with

one chord and K1,7 is demonstrated in the following Figure 6.2.

u3

u4 u5

u6

u7

u1

u2

v3 v4 v5 v6 v7v1 v2

5

7

3

1

9

2 4 6 8 10 12 14

11

13

Figure 6.2: SDC labeling in the graph constructed from ringsum of C7 with one chord and K1,7

Theorem 6.1.3. Cn,3 ⊕K1,n is a sum DC graph.



n}⋃{u2un, u2un−1}, where u2un and u2un−1 are the chords of Cn.


Consider a bijection f from V (Cn,3 ⊕K1,n) to {1, 2, 3, . . . , 2n} defined as below.

f(uj) = 2j − 1; 1 ≤ j ≤ n− 3.

f(un−2) = 2n.

f(uj) = 2j − 3; n− 1 ≤ j ≤ n.

f(vk) = 2k; 1 ≤ k ≤ n− 1.

f(vn) = 2n− 1.

As per this pattern, allocate the vertices such that for any edge uiui+1 ∈ E(Cn,3 ⊕K1,n),

f(uj) | f(uj+1), 1 ≤ j ≤ n− 4.

Also f(u1) - f(vk), 1 ≤ k ≤ n− 1.

121



ef (0) = n+ 1 = ef (1).

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Cn,3 ⊕K1,n is SDC graph.

Example 6.1.3. SDC labeling in the graph C7,3 ⊕K1,7 is demonstrated in the fol-

lowing Figure 6.3.

2

11

u3

u4 u5

u6

u7

u1

u2

v3 v4 v5 v6 v7v1 v2

3

5

7

9

13

1

4 6 8 10 12

14

Figure 6.3: SDC labeling in C7,3 ⊕K1,7

Theorem 6.1.4. Cn(1, 1, n− 5)⊕K1,n is SDC graph.

Proof. Let G be the cycle with triangle Cn(1, 1, n− 5).



Let E(G⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤ n}⋃{u1u3,

u3un−1, un−1u1}, where u1, u3 and un−1 are the vertices of the triangle formed by the

chords u1u3, u3un−1 and u1un−1.


Consider a bijection f from V (G⊕K1,n) to {1, 2, 3, . . . , 2n} defined as below.

f(u1) = 1.

f(uj) = 2j − 1; 2 ≤ j ≤ n− 1.

122


f(un) = 2n.

f(vk) = 2k; 1 ≤ k ≤ n− 1.

f(vn) = 2n− 1.

As per this pattern, allocate the vertices such that for any edge uiui+1 ∈ E(G⊕K1,n),

f(uj) | f(uj+1), 1 ≤ j ≤ n− 2.

f(u1) - f(vk), 1 ≤ k ≤ n− 1

f(u1) | f(u3), f(u3) | f(un−1), f(un−1) | f(u1).


ef (0) = n+ 1, ef (1) = n+ 2.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Cn(1, 1, n− 5)⊕K1,n is SDC graph.

Example 6.1.4. SDC labeling in the graph C8(1, 1, 3)⊕K1,8 is demonstrated in the


v3 v4 v5 v6 v7v1 v2

u3

u4

u5

u6

u7

u1

v8

u2 u8

2

1

5

7

9

11

13

15

3

4 6 8 10 12

16

14

Figure 6.4: SDC labeling in C8(1, 1, 3)⊕K1,8

Theorem 6.1.5. Wn ⊕K1,n is SDC graph.

Proof. Let V (Wn ⊕K1,n) = {u0, uj, vj | 1 ≤ j ≤ n}, where V (Wn) = {u0, uj | 1 ≤j ≤ n} and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.

123


Here u0 is apex vertex, uj(1 ≤ j ≤ n) are rim vertices of Wn and vj(1 ≤ j ≤ n) are

the pendant vertices, u1 is apex vertex of star graph.

Let E(Wn⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n−1}⋃{unu1}⋃{u0uj | 1 ≤ j ≤ n}⋃{u1vj |


Consider a bijection f from V (Wn ⊕K1,n) to {1, 2, 3, . . . , 2n+ 1} defined as below.

f(u0) = 2.

f(u1) = 1.

f(uj) =

j ; j ≡ 0(mod 4)

j + 1 ; j ≡ 1, 2(mod 4)

j + 2 ; j ≡ 3(mod 4) 2 ≤ j ≤ n.

f(vk) = f(un) + 1 + k; 1 ≤ k ≤ n.

By looking into the above prescribed pattern

ef (1) =

⌈3n

2

⌉, ef (0) =

⌊3n

2

⌋.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Wn ⊕K1,n is SDC graph.

Example 6.1.5. SDC labeling in the graph W6 ⊕ K1,6 is demonstrated in the fol-

lowing Figure 6.5.

9

5

7

11 13

1

3

4

6

2

8 10 12

u0

v6

u3

u4

u5

u1

u2

v1 v2 v4v3 v5

u6

Figure 6.5: SDC labeling in W6 ⊕K1,6

124


Theorem 6.1.6. Fln ⊕K1,n is SDC graph.

Proof. Let V (Fln⊕K1,n) = {u0, uj, vj, wj | 1 ≤ j ≤ n}, where V (Fln) = {u0, uj, wj |1 ≤ j ≤ n} and V (K1,n) = {w1, vj | 1 ≤ j ≤ n}.Here u0 is apex vertex, uj(1 ≤ j ≤ n) are internal vertices and wj(1 ≤ j ≤ n) are

external vertices of Fln and d(vj) = 1, where 1 ≤ j ≤ n, w1 is apex vertex of star

graph.

Let E(Fln ⊕ K1,n) = {ujuj+1 | 1 ≤ j ≤ n − 1}⋃{unu1}⋃{u0uj | 1 ≤ j ≤

n}⋃{u0wj | 1 ≤ j ≤ n}⋃{ujwj | 1 ≤ j ≤ n}⋃{w1vj | 1 ≤ j ≤ n}.It is to be noted that, |V (Fln ⊕K1,n)| = 3n+ 1 and |E(Fln ⊕K1,n)| = 5n.

Consider a bijection f from V (Fln ⊕K1,n) to {1, 2, 3, . . . , 3n+ 1} defined as below.

f(u0) = 1.

f(uj) = 2j + 1; 1 ≤ j ≤ n.

f(wj) = 2j; 1 ≤ j ≤ n.

f(vk) = f(un) + k; 1 ≤ k ≤ n.

As per this pattern, allocate the vertices such that for any edge ujuj+1 ∈ E(Fln ⊕K1,n),

f(uj) | f(uj+1), 1 ≤ j ≤ n− 1.

Further f(u0) | f(uj), f(u0) - f(wj), 1 ≤ j ≤ n

and f(w1) | f(vk) whenever k is odd, 1 ≤ k ≤ n.


ef (1) =

⌈5n

2

⌉, ef (0) =

⌊5n

2

⌋

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Fln ⊕K1,n is SDC graph.

Example 6.1.6. SDC labeling in the graph Fl4 ⊕ K1,4 is demonstrated in the fol-

lowing Figure 6.6.

125


3

2

9 854

7

6

11

1310

12

1

u1

u2

u3

u4

w1

w2

w3

u0

v1

v2 v3

v4

w4

Figure 6.6: SDC labeling in Fl4 ⊕K1,4

Theorem 6.1.7. Gn ⊕K1,n is SDC graph.

Proof. Let V (Gn ⊕ K1,n) = {u0, uj, vk | 1 ≤ j ≤ 2n, 1 ≤ k ≤ n}, where V (Gn) =

{u0, uj | 1 ≤ j ≤ 2n} and V (K1,n) = {u1, vk | 1 ≤ k ≤ n}.Here u0 is apex vertex, d(u2i−1) = 3(1 ≤ i ≤ n) and d(u2i) = 2(1 ≤ i ≤ n) and

d(vk) = 1, where 1 ≤ k ≤ n, u1 is apex vertex of star graph.

Let E(Gn ⊕ K1,n) = {ujuj+1 | 1 ≤ j ≤ 2n − 1}⋃{u2nu1}⋃{u0u2j−1 | 1 ≤ j ≤

n}⋃{u1vk | 1 ≤ k ≤ n}.It is to be noted that, |V (Gn ⊕K1,n)| = 3n+ 1 and |E(Gn ⊕K1,n)| = 4n.

Consider a bijection f from V (Gn ⊕K1,n) to {1, 2, 3, . . . , 3n+ 1} define as belows.

Case:1 n ≡ 1, 3(mod 4)

f(u0) = 2.

f(u1) = 1.

f(uj) =

j + 1 ; j ≡ 1, 2(mod 4)

j + 2 ; j ≡ 3(mod 4)

j ; j ≡ 0(mod 4) 2 ≤ j ≤ n.

f(vk) = f(un−1) + k; 1 ≤ k ≤ n.

126


Case:2 n ≡ 2, 4(mod 4)

f(u0) = 2.

f(u1) = 1.

f(uj) =

j + 1 ; j ≡ 1, 2(mod 4)

j + 2 ; j ≡ 3(mod 4)

j ; j ≡ 0(mod 4) 2 ≤ j ≤ n− 1.

f(un) = 3n+ 1.

f(vk) = f(un−1) + k; 1 ≤ k ≤ n− 1.

f(vn) = 2n.

By looking into the above prescribed pattern, ef (1) = ef (0) = 2n.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Gn ⊕K1,n is SDC graph.

Example 6.1.7. SDC labeling in the graph G6⊕K1,6 is demonstrated in the following

Figure 6.7.

u0

u2

u3

u4

u5

u6

u7

u8

u9

u10

u12

u1

u11

7

v1 v2 v3 v4 v5 v6

1

2

35

4

6

98

10

11

13

19

18171614 15 12

Figure 6.7: SDC labeling in G6 ⊕K1,6

Theorem 6.1.8. Pn ⊕K1,n is SDC graph.


127



Consider a bijection f from V (Pn ⊕K1,n) to {1, 2, 3, . . . , 2n} defined as below.

f(uj) = 2j; 1 ≤ j ≤ n.

f(vk) = 2k − 1; 1 ≤ k ≤ n.

As per this pattern, allocate the vertices such that for any edge uiui+1 ∈ E(Pn ⊕K1,n),

f(uj) | f(uj+1), 1 ≤ j ≤ n− 1.

Also f(u1) - f(vk) 1 ≤ k ≤ n.


ef (0) = n, ef (1) = n− 1.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Pn ⊕K1,n is SDC graph.

Example 6.1.8. SDC labeling in the graph P5⊕K1,5 is demonstrated in the following

Figure 6.8.

246810u1u2u3u4u5

v1

v2

v3

v4

v5

5

7

31

9

Figure 6.8: SDC labeling in P5 ⊕K1,5

Theorem 6.1.9. Sn ⊕K1,n is SDC graph.

Proof. Let V (Sn ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (Sn) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of Sn as well as of star graph.

Let E(Sn⊕K1,n) = {ujuj+1, unu1 | 1 ≤ j ≤ n−1}⋃{u1uj | 2 ≤ j ≤ n−1}⋃{u1vj |

128


1 ≤ j ≤ n}.It is to be noted that, |V (Sn ⊕K1,n)| = 2n and |E(Sn ⊕K1,n)| = 3n− 3.

Consider a bijection f from V (Sn ⊕K1,n) to {1, 2, 3, . . . , 2n} defined as below.

f(u1) = 1.

f(uj) =

j ; j ≡ 1, 2(mod 4)

j + 1 ; j ≡ 3(mod 4)

j − 1 ; j ≡ 0(mod 4) 2 ≤ j ≤ n.

To label the vertices {vj | 1 ≤ j ≤ n}, let us consider the below possibilities.

Case:1 n ≡ 0, 1, 2(mod 4).

f(vk) = n+ k; 1 ≤ k ≤ n.

Case:2 n ≡ 3(mod 4).

f(v1) = n

f(vk) = n+ k; 2 ≤ k ≤ n.

The edge label conditions constructed due to the above labeling pattern is shown in

the below table.

Cases of n Edge label conditions

n ≡ 0, 2(mod 4) ef (0) =⌈3n−3

2

⌉, ef (1) =

⌊3n−3

2

⌋

n ≡ 1, 3(mod 4) ef (1) = 3n−32

= ef (0)

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Sn ⊕K1,n is SDC graph.

Example 6.1.9. SDC labeling in the graph S7⊕K1,7 is demonstrated in the following

Figure 6.9.

129


u3

u4u5

u6

u7

u1

u2

v3 v4 v5 v6 v7v1 v2

3 5

7 9 13

1

4 6

8

10 12 14

2

11

Figure 6.9: DC labeling in S7 ⊕K1,7

Theorem 6.1.10. DFn ⊕K1,n is SDC graph.

Proof. Let V (DFn ⊕K1,n) = {u,w, uj, vj | 1 ≤ j ≤ n}, where V (DFn) = {u,w, uj |1 ≤ j ≤ n} and V (K1,n) = {u, vj | 1 ≤ j ≤ n}.Here u,w are apex vertices of DFn, d(uj) = 3(1 ≤ j ≤ n), d(vj) = 1(1 ≤ j ≤ n) and

u is apex vertex of star graph.


Consider a bijectionf from V (DFn⊕K1,n) to {1, 2, 3, . . . , 2n+ 2} defined as below.

f(w) = 2.

f(u) = 1.

f(uj) =

j + 2 ; j ≡ 0, 1(mod 4)

j + 3 ; j ≡ 2(mod 4)

j + 1 ; j ≡ 3(mod 4) 1 ≤ j ≤ n.

To label the vertices {vj | 1 ≤ j ≤ n}, let us consider the below possibilities.

Case:1 n ≡ 0, 1, 3(mod 4).

f(vk) = f(un) + k; 1 ≤ k ≤ n.

130


Case:2 n ≡ 2(mod 4).

f(v1) = n+ 2.

f(vk) = f(un) + k; 2 ≤ k ≤ n.


the below table.


n ≡ 0, 2(mod 4) ef (0) = 2n, ef (1) = 2n− 1

n ≡ 1, 3(mod 4) ef (1) = 2n, ef (0) = 2n− 1

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, DFn ⊕K1,n is SDC graph.

Example 6.1.10. SDC labeling in the graph DF5 ⊕ K1,5 is demonstrated in the


u1 u2 u3 u4 u5

7

9

v1

v2

v3

v4

v5

u

w

1

3 5

11

2

4 6

8

10

12

v1

v2

v3

v4

v5

u

w

1

3 5

11

2

4 6

8

10

12

u1 u2 u3 u4 u5

7

9

Figure 6.10: SDC labeling in DF5 ⊕K1,5

Theorem 6.1.11. K2,n ⊕K1,n is SDC graph.


Let E(K2,n ⊕K1,n) = {uuj | 1 ≤ j ≤ n}⋃{wuj | 1 ≤ j ≤ n}⋃{u1vj | 1 ≤ j ≤ n}.

131


It is to be noted that, |V (K2,n ⊕K1,n)| = 2n+ 2 and |E(K2,n ⊕K1,n)| = 3n.

Consider a bijection f from V (K2,n⊕K1,n) to {1, 2, 3, . . . , 2n+ 2} defined as below.

f(u) = 1.

f(w) = 2.

f(uj) = j + 2; 1 ≤ j ≤ n.

f(vk) = f(un) + k; 1 ≤ k ≤ n.


the below table.


n ≡ 0, 2(mod 4) ef (0) = 3n2

= ef (1)

n ≡ 1, 3(mod 4) ef (0) =⌈3n2

⌉, ef (1) =

⌊3n2

⌋

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, K2,n ⊕K1,n is SDC graph.

Example 6.1.11. SDC labeling in the graph K2,7 ⊕ K1,7 is demonstrated in the


u1

u2

v1

v2

u3

v3 v4

v5

u4

w

u

u5

u6

u7

v6

v7

1 5

7

1113

9

15

3

2

4

6

8

10

1214

16

Figure 6.11: SDC labeling in K2,7 ⊕K1,7

132

6.2. SDC Labeling in the Graphs constructed from Corona Product with K1

6.2 SDC Labeling in the Graphs constructed from Corona

Product with K1

Kanani and Bosmia[25] illuminated and derived some fascinating graphs by consid-

ering corona product of K1 with different graph families for divisor cordial labeling.

Under the inspiration of this credibility, in the current segment we demonstrate some

new graphs constructed from the graph operation corona product for SDC labeling.

Theorem 6.2.1. K1,n �K1 is SDC graph.

Proof. Let V (K1,n�K1) = {vj, v′j | 0 ≤ j ≤ n}, where v0 is apex vertex, vj(1 ≤ j ≤n) are pendant vertices of K1,n and v′j(0 ≤ j ≤ n) are the lately inserted vertices to

construct the graph K1,n �K1.

Let E(K1,n �K1) = {v0vj; 1 ≤ j ≤ n}⋃{vjv′j; 0 ≤ j ≤ n}.Also it is to be noted that, |V (K1,n �K1)| = 2n+ 2 and |E(K1,n �K1)| = 2n+ 1.

Consider a bijection f : V (K1,n �K1)→ {1, 2, 3, . . . , 2n+ 2} defined as below.

f(v0) = 1.

f(v′0) = 2.

f(vj) = 2j + 1 1 ≤ j ≤ n.

f(v′j) = 2j + 2 1 ≤ j ≤ n.


ef (1) =

⌊2n+ 1

2

⌋, ef (0) =

⌈2n+ 1

2

⌉.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, K1,n �K1 is SDC graph.

Example 6.2.1. SDC labeling in the graph K1,6 � K1 is demonstrated in the fol-

lowing Figure 6.12.

133


v1 v2 v3 v4 v5 v6

v2' v3' v4' v5' v6'

v0'

v01

6

3

4 8 1210 14

5 1197 13

2

v1'

Figure 6.12: SDC labeling in K1,6 �K1

Theorem 6.2.2. K2,n �K1 is sum divisor cordial graph for n ≡ 1, 2, 3(mod 4).

Proof. Let V (K2,n � K1) = {uj, u′j | 1 ≤ j ≤ 2}⋃{vj, v′j, | 1 ≤ j ≤ n}, where

V (K2,n) = {u1, u2, vj | 1 ≤ j ≤ n} and u′1, u′2, v′j(1 ≤ j ≤ n) are lately inserted

vertices to construct the graph K2,n �K1.

Let E(K2,n �K1) = {u1vj, u2vj | 1 ≤ j ≤ n}⋃{u1u′1, u2u′2}⋃{vjv′j; 1 ≤ j ≤ n}.

Also it is to be noted that, |V (K2,n �K1)| = 2n+ 4 and |E(K2,n �K1)| = 3n+ 2.


f(u1) = 1.

f(u2) = 2.

f(vj) = j + 2; 1 ≤ j ≤ n.

Case 1: n ≡ 1(mod 4).

f(u′1) = 2n+ 3.

f(u′2) = 2n+ 4.

f(v′2j−1) = (n+ 2) + 2j 1 ≤ j ≤ k.

f(v′2j) = (n+ 2) + (2j − 1); 1 ≤ j ≤ k.

f(v′j) = (n+ 2) + j; 2k + 1 ≤ j ≤ n,where k =n− 1

4.

Case 2: n ≡ 2(mod 4).

f(u′1) = 2n+ 3.

134


f(u′2) = 2n+ 4.

f(v′2j−1) = (n+ 2) + (2j); 1 ≤ j ≤ k.

f(v′2j) = (n+ 2) + (2j − 1); 1 ≤ j ≤ k.

f(v′j) = (n+ 2) + j; 2k + 1 ≤ j ≤ n,where k =n+ 2

4.

Case 3: n ≡ 3(mod 4).

f(u′1) = 2n+ 4.

f(u′2) = 2n+ 3.

f(v′2j−1) = (n+ 2) + 2j 1 ≤ j ≤ k.

f(v′2j) = (n+ 1) + (2j − 1); 1 ≤ j ≤ k.

f(v′j) = (n+ 2) + j; 2k + 1 ≤ j ≤ n,where k =n+ 1

4.

By looking into the above prescribed pattern, the below table describes edge label

conditions.


n ≡ 1(mod 4) ef (1) =⌈3n+2

2

⌉, ef (0) =

⌊3n+2

2

⌋

n ≡ 2(mod 4) ef (1) = 3n+22

= ef (0)

n ≡ 3(mod 4) ef (1) =⌊3n+2

2

⌋, ef (0) =

⌈3n+2

2

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.



lowing Figure 6.13.

u1'

u2

u2'

u1

3 5

119

7

12

1 2

8

64

10

13 14

v2' v3' v4' v5'v1'

v1 v2 v3 v4 v5


135


Theorem 6.2.3. K3,n �K1 is sum divisor cordial graph.

Proof. Let V (K3,n � K1) = {uj, u′j | 1 ≤ j ≤ 3}⋃{vj, v′j | 1 ≤ j ≤ n}, where

V (K3,n) = {u1, u2, u3, vj | 1 ≤ j ≤ n} and u′1, u′2, u′3, v′j(1 ≤ j ≤ n) are the lately

inserted vertices to construct the graph K3,n �K1.

Let E(K3,n �K1) = {uju′j | 1 ≤ j ≤ 3}⋃{vjv′j | 1 ≤ j ≤ n}⋃{u1vj, u2vj, u3vj | 1 ≤j ≤ n}.Also it is to be noted that, |V (K3,n �K1)| = 2n+ 6 and |E(K3,n �K1)| = 4n+ 3.


f(uj) = j 1 ≤ j ≤ 3.

f(u′1) = 5.

f(u′2) = 4.

f(u′3) = 6.

f(vj) = 6 + (2j − 1). 1 ≤ j ≤ n;

f(v′j) = 6 + (2j); 1 ≤ j ≤ n.


ef (1) =

⌈4n+ 3

2

⌉, ef (0) =

⌊4n+ 3

2

⌋.

Then we get, |ef (0)− ef (1)| ≤ 1.



lowing Figure 6.14.

u1' u2' u3'64

8 12 14

1 3

5

1197 13

2

1715 19

1816 2010

v2' v3' v4' v5' v6'v1' v7'

v1 v2 v3 v6 v7v4 v5

u1 u2 u3


136


Theorem 6.2.4. Wn �K1 is sum divisor cordial graph.

Proof. Let V (Wn � K1) = {vj | 0 ≤ j ≤ n}⋃{v′j | 0 ≤ j ≤ n}, where v0 is apex

vertex and vj(1 ≤ j ≤ n) are rim vertices of Wn.

Let v′j(0 ≤ j ≤ n) be the lately inserted vertices to construct the graph Wn �K1.

Let E(Wn �K1) = {v0vj | 1 ≤ j ≤ n}⋃{vjvj+1 | 1 ≤ j ≤ n− 1}⋃{vnv1}⋃{vjv′j |

0 ≤ j ≤ n}.Also it is to be noted that, |V (Wn �K1)| = 2n+ 2 and |E(Wn �K1)| = 3n+ 1.

Consider a bijection f : V (Wn �K1)→ {1, 2, 3, . . . , 2n+ 2} defined as below.

Case 1: n ≡ 0, 2(mod 4).

f(v0) = 1.

f(v′0) = 2n+ 2.

f(v2j−1) = 4j − 2; 1 ≤ j ≤ n

2.

f(v2j) = 4j − 1; 1 ≤ j ≤ n

2.

f(v′2j−1) = 4j; 1 ≤ j ≤ n

2.

f(v′2j) = 4j + 1; 1 ≤ j ≤ n

2.

Case 2: n ≡ 1, 3(mod 4).

f(v0) = 1.

f(v′0) = 2n+ 1.

f(v2j−1) = 4j − 2; 1 ≤ j ≤ n+ 1

2.

f(v2j) = 4j − 1; 1 ≤ j ≤ n− 1

2.

f(v′2j−1) = 4j; 1 ≤ j ≤ n+ 1

2.

f(v′2j) = 4j + 1; 1 ≤ j ≤ n− 1

2.


conditions.

137



n ≡ 0, 2(mod 4) ef (1) =⌊3n+1

2

⌋, ef (0) =

⌈3n+1

2

⌉

n ≡ 1, 3(mod 4) ef (1) = 3n+12

= ef (0)

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Wn �K1 is SDC graph.

Example 6.2.4. SDC labeling in the graph W7�K1 is demonstrated in the following

Figure 6.15.

v0

v1

v7

v6

v5

v4

v7'

v1'

v6'

v5'

v4'

v3

v2

v3'

v2'

v0'

1

54

32

9

8

7

6

13

1211

10

15

16

14

Figure 6.15: SDC labeling in W7 �K1

Theorem 6.2.5. Hn �K1 is sum divisor cordial graph.

Proof. Let V (Hn �K1) = {v0, vj, uj | 1 ≤ j ≤ n}⋃{v′0, v′j, u′j | 1 ≤ j ≤ n}, where

v0 is apex vertex, vj, uj are vertices of the helm Hn and d(vj) = 4(1 ≤ j ≤ n),

d(uj) = 1(1 ≤ j ≤ n).

Let v′0, v′1, v′2, . . . , v

′n, u

′1, u′2, . . . , u

′n be the lately inserted vertices to construct the

graph Hn �K1.

Let E(Hn � K1) = {v0vj, viuj | 1 ≤ j ≤ n}⋃{vjvj+1 | 1 ≤ j ≤ n}⋃{vnv1}⋃

{v0v′0, viv′j, uju′j | 1 ≤ j ≤ n}.Also it is to be noted that, |V (Hn �K1)| = 4n+ 2 and |E(Hn �K1)| = 5n+ 1.

Consider a bijection f : V (Hn �K1)→ {1, 2, 3, . . . , 4n+ 2} defined as below.

Case 1: n ≡ 0, 1(mod 4).

f(v0) = 1.

f(v′0) = 4n+ 2.

138


f(vj) =

j + 1 j ≡ 0, 1(mod 4).

j + 2 j ≡ 2(mod 4).

j j ≡ 3(mod 4); 1 ≤ j ≤ n.

f(v′j) = (n+ 1) + 2j; 1 ≤ j ≤ n.

f(uj) = n+ 2j; 1 ≤ j ≤ n.

f(u′j) = (3n+ 1) + i; 1 ≤ j ≤ n.

Case 2: n ≡ 2(mod 4).

f(vj) =

j j ≡ 0, 1(mod 4).

j + 1 j ≡ 2(mod 4).

j − 1 j ≡ 3(mod 4); 1 ≤ j ≤ n.

f(v′j) = (n+ 1) + 2j; 2 ≤ j ≤ n.

f(v′1) = n+ 2.

f(u1) = n;

f(uj) = n+ 2j; 2 ≤ j ≤ n.

f(u′j) = (3n+ 1) + j; 1 ≤ j ≤ n.

Case 3: n ≡ 3(mod 4).

f(vj) =

j + 1 j ≡ 0, 1(mod 4)

j + 2 j ≡ 2(mod 4)

j j ≡ 3(mod 4); 1 ≤ j ≤ n.

f(v0) = 4n+ 2.

f(v′0) = 1.

f(v′j) = (n+ 1) + 2j; 1 ≤ j ≤ n.

f(uj) = n+ 2j; 1 ≤ j ≤ n.

f(u′j) = (3n+ 1) + j; 1 ≤ j ≤ n.


139


conditions.


n ≡ 0, 2(mod 4) ef (1) =⌊5n+1

2

⌋, ef (0) =

⌈5n+1

2

⌉

n ≡ 1, 3(mod 4) ef (1) = 5n+12

= ef (0)

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Hn �K1 is SDC graph.

Example 6.2.5. SDC labeling in the graph H7�K1 is demonstrated in the following

Figure 6.16.

u1u2

u2'

v0

v1

v'1

v7

v7'

v6

v6'v5

v5'

v4v4'

v3

v3'v2

v'2

u7

u7'

u1'

u6

u5

u5'

u4 u6'

u4'

u3u3'

v0'1

3

24

5

7

6

8

910

21

22

20

19

17

13

18

15

12

16

14

11

23

27

29

28

26

25

24

30

Figure 6.16: SDC labeling in H7 �K1

Theorem 6.2.6. Fln �K1 is sum divisor cordial graph.

Proof. Let V (Fln �K1) = {v0, vj, uj, | 1 ≤ j ≤ n}⋃{v′0, v′j, u′j, | 1 ≤ j ≤ n}, where

v0 is apex vertex, vj, uj are the vertices in the flower graph Fln and d(vj) = 4(1 ≤j ≤ n), d(uj) = 2(1 ≤ j ≤ n) .

Let v′0, v′1, v′2, . . . , v

′n, u

′1, u′2, . . . , u

′n be the lately inserted vertices to construct the

graph Fln �K1.

Let E(Fln � K1) = {v0vj, v0uj, viuj | 1 ≤ j ≤ n.}⋃{vjvj+1 | 1 ≤ j ≤ n −1}⋃{vnv1}

⋃ {vjv′j, uju′j | 1 ≤ j ≤ n}⋃{v0v′0}.Also it is to be noted that, |V (Fln �K1)| = 4n+ 2 and |E(Fln �K1)| = 6n+ 1.

140


Consider a bijection f : V (Fln �K1)→ {1, 2, 3, . . . , 4n+ 2} defined as below.

f(vj) =

j + 1 j ≡ 0, 1(mod 4).

j + 2 j ≡ 2(mod 4).

j j ≡ 3(mod 4); 1 ≤ j ≤ n.

f(v0) = 1.

f(v′0) = 4n+ 2.

Case 1: n ≡ 0(mod 4).

f(v′4j−3) = 3n+ 4j − 1; 1 ≤ j ≤ n

4.

f(v′4j−2) = 3n+ 4j + 1; 1 ≤ j ≤ n

4.

f(v′4j−1) = 3n+ 4j − 2; 1 ≤ j ≤ n

4.

f(v′4j) = 3n+ 4j; 1 ≤ j ≤ n

4.

f(u2j−1) = n+ 4j − 2; 1 ≤ j ≤ n

2.

f(u2j) = n+ 4j − 1; 1 ≤ j ≤ n

2.

f(u′2j−1) = n+ 4j; 1 ≤ j ≤ n

2.

f(u′2j) = n+ 4j + 1; 1 ≤ j ≤ n

2.

Case 2: n ≡ 2(mod 4).

f(v′4j−3) = 3n+ 4j − 3; 1 ≤ j ≤ n+ 2

4.

f(v′4j−2) = 3n+ 4j − 1; 1 ≤ j ≤ n+ 2

4.

f(v′4j−1) = 3n+ 4j; 1 ≤ j ≤ n− 2

4.

f(v′4j) = 3n+ 4j + 2; 1 ≤ j ≤ n− 2

4.

f(u2j−1) = n+ 4j − 3; 1 ≤ j ≤ n

2.

f(u2j) = n+ 4j; 1 ≤ j ≤ n

2.

f(u′2j−1) = n+ 4j − 1; 1 ≤ j ≤ n

2.

f(u′2j) = n+ 4j + 2; 1 ≤ j ≤ n

2.

141


Case 3: n ≡ 1(mod 4).

f(v′4j−3) = 3n+ 4j; 1 ≤ j ≤ n− 1

4.

f(v′4j−2) = 3n+ 4j + 2; 1 ≤ j ≤ n− 1

4.

f(v′4j−1) = 3n+ 4j − 3; 1 ≤ j ≤ n− 1

4.

f(v′4j) = 3n+ 4j − 1; 1 ≤ j ≤ n− 1

4.

f(u2j−1) = n+ 4j − 2; 1 ≤ j ≤ n+ 1

2.

f(u′2j−1) = n+ 4j; 1 ≤ j ≤ n+ 1

2.

f(u2j) = n+ 4j − 1; 1 ≤ j ≤ n− 1

2.

f(u′2j) = n+ 4j + 1; 1 ≤ j ≤ n− 1

2.

f(v′n) = 4n.

Case 4: n ≡ 3(mod 4).

f(v′4j−3) = 3n+ 4j; 1 ≤ j ≤ n+ 1

4.

f(v′4j−2) = 3n+ 4j + 2; 1 ≤ j ≤ n− 3

4.

f(v′4j−1) = 3n+ 4j − 3; 1 ≤ j ≤ n− 3

4.

f(v′4j) = 3n+ 4j − 1; 1 ≤ j ≤ n− 3

4.

f(u2j−1) = n+ 4j − 2; 1 ≤ j ≤ n+ 1

2.

f(u′2j−1) = n+ 4j; 1 ≤ j ≤ n+ 1

2.

f(u2j) = n+ 4j − 1; 1 ≤ j ≤ n− 1

2.

f(u′2j) = n+ 4j + 1; 1 ≤ j ≤ n− 1

2.

f(v′n−1) = 4n− 2.

f(v′n) = 4n.

142



ef (1) =

⌈6n+ 1

2

⌉, ef (0) =

⌊6n+ 1

2

⌋.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Fln �K1 is SDC graph.

Example 6.2.6. SDC labeling in the graph Fl7�K1 is demonstrated in the following

Figure 6.17.

v0

v1

v1'

v7

v7'

v6

v6'v5

v5'

v4v4'

v3

v3'v2

v2'

u1'

u1

u7u7'

u6

u6'

u5

u5'

u4

u4'

u3' u3

u2

u2'

v0'5

7

23

21

17

1

8

6

11

4

18

2

14

20

16

13

19

15 3

912

28

10

29

26

2527

22

24

30

Figure 6.17: SDC labeling in Fl7 �K1

Theorem 6.2.7. Fn �K1 is SDC graph.

Proof. Let V (Fn � K1) = {vj, v′j | 1 ≤ j ≤ n}⋃{u1, u′1}, where u1 is apex vertex,

vj(1 ≤ j ≤ n) are the vertices of the path Pn corresponding to the fan graph Fn and

let u′1, v′1, v′2, . . . , v

′n are lately inserted vertices to construct the graph Fn �K1.

Let E(Fn �K1) = {u1vj | 1 ≤ j ≤ n}⋃{vjvj+1 | 1 ≤ j ≤ n − 1}⋃{viv′j | 1 ≤ j ≤n}⋃{u1u′1}.Also it is to be noted that, |V (Fn �K1)| = 2n+ 2 and |E(Fn �K1)| = 3n.

Consider a bijection f : V (Fn �K1)→ {1, 2, 3, . . . , 2n+ 2} defined as below.

143


Case 1: n ≡ 0, 3(mod 4).

f(vj) =

j + 1 j ≡ 0, 1(mod 4).

j + 2 j ≡ 2(mod 4).

j j ≡ 3(mod 4); 1 ≤ j ≤ n.

f(u1) = 1.

f(u′1) = 2n+ 2.

f(v′j) = (n+ 1) + j; 1 ≤ j ≤ n.

Case 2: n ≡ 1(mod 4).

f(vj) =

j + 1 j ≡ 0, 1(mod 4);

j + 2 j ≡ 2(mod 4);

j j ≡ 3(mod 4); 1 ≤ j ≤ n.

f(u1) = 1.

f(u′1) = 2n+ 1.

f(v′j) = (n+ 1) + j; 1 ≤ j ≤ n− 1.

f(v′n) = 2n+ 2.

Case 3: n ≡ 2(mod 4).

f(vj) =

j + 1 j ≡ 0, 1(mod 4);

j + 2 j ≡ 2(mod 4);

j j ≡ 3(mod 4); 1 ≤ j ≤ n.

f(u1) = 1.

f(u′1) = 2n+ 1.

f(v′j) = n+ 1 + j; 2 ≤ j ≤ n− 1.

f(v′1) = n+ 1.

f(v′n) = 2n+ 2.

144




n ≡ 1(mod 4) ef (1) =⌈3n2

⌋, ef (0) =

⌊3n2

⌉

n ≡ 3(mod 4) ef (1) =⌊3n2

⌋, ef (0) =

⌈3n2

⌉

n ≡ 0, 2(mod 4) ef (1) = 3n2

= ef (0)

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Fn �K1 is SDC graph.

Example 6.2.7. SDC labeling in the graph F8�K1 is demonstrated in the following

Figure 6.18.

u1'

u1

64 8

1210 14

1

3 5

11

97

13

2

1715

18

16

v2' v3' v4' v5' v6'v1' v7'

v1 v2 v3 v6 v7v4 v5

v8'

v8

Figure 6.18: SDC labeling in F8 �K1

Theorem 6.2.8. DFn �K1 is SDC graph.

Proof. Let V (DFn � K1) = {vj, v′j, | 1 ≤ j ≤ n}⋃{uj, u′j | 1 ≤ j ≤ 2}, where

u1, u2 are apex vertices of degree n and vj(1 ≤ j ≤ n) are the vertices of path Pn

corresponding to the double fan DFn.

Let u′1, u′2, v′1, v′2, . . . , v

′n be the lately inserted vertices to construct the graph DFn�

K1.

Let E(DFn � K1) = {u1vj, u2vj | 1 ≤ j ≤ n}⋃{vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vjv′j |1 ≤ j ≤ n.}⋃{u1u′1}

⋃{u2u′2}.Also it is to be noted that, |V (DFn �K1)| = 2n+ 4 and |E(DFn �K1)| = 4n+ 1.

Consider a bijection f : V (DFn �K1)→ {1, 2, 3, . . . , 2n+ 4} defined as below.

145


Case 1: n ≡ 0, 2(mod 4).

f(u1) = 1.

f(u′1) = 2n+ 4.

f(u2) = 2.

f(u′2) = 2n+ 3.

f(vj) = 2 + j; 1 ≤ j ≤ n.

f(v′j) = n+ 2 + j; 1 ≤ j ≤ n.

Case 2: n ≡ 1, 3(mod 4).

f(u1) = 1.

f(u′1) = 2n+ 2.

f(u2) = 2.

f(u′2) = 2n+ 4.

f(vj) = 2 + j; 1 ≤ j ≤ n.

f(v′2j−1) = n+ (2j − 1) + 3; 1 ≤ j ≤ n+ 1

2.

f(v′2j) = n+ 2j + 1; 1 ≤ j ≤ n− 1

2.


conditions.


n ≡ 1, 3(mod 4) ef (1) =⌈4n+1

2

⌉, ef (0) =

⌊4n+1

2

⌋

n ≡ 0, 2(mod 4) ef (1) =⌊4n+1

2

⌋, ef (0) =

⌈4n+1

2

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, DFn �K1 is SDC graph.

Example 6.2.8. SDC labeling in the graph DF6 � K1 is demonstrated in the fol-

lowing Figure 6.19.

146


u2

u2'

3 4 5 6 7 8

1

2

16

15

9 10 11 12 13 14v2' v3' v4' v5' v6'v1'

v1 v2 v3 v6v4 v5

u1

u1'

Figure 6.19: SDC labeling in DF6 �K1

Theorem 6.2.9. S(K1,n)�Kn is SDC graph.

Proof. Let V (S(K1,n) � K1) = {v0, v′0}⋃{vj, uj, v′j, u′j | 1 ≤ j ≤ n}, where v0 is

apex vertex, vj, uj are the vertices of the graph S(K1,n) and d(vj) = 2(1 ≤ j ≤ n),

d(uj) = 1(1 ≤ j ≤ n) .

Let v′0, v′j, u′j (1 ≤ i ≤ n) be the lately inserted vertices to construct the graph

S(K1,n) �K1.

Let E(S(K1,n)�K1) = {v0vj, vjuj | 1 ≤ j ≤ n.}⋃{v0v′0, viv′j, uju′j | 1 ≤ j ≤ n.}Also it is to be noted that, |V (S(K1,n)�K1)| = 4n+2 and |E(S(K1,n)�K1)| = 4n+1.

Consider a bijection f : V (S(K1,n)�K1)→ {1, 2, 3, . . . , 4n+ 2} defined as below.

f(v0) = 1.

f(v′0) = 4n+ 2.

f(vj) = 2j + 1; 1 ≤ j ≤ n.

f(v′j) = 2j; 1 ≤ j ≤ n.

f(uj) = (2n+ 1) + 2j; 1 ≤ j ≤ n.

f(u′j) = 2n+ 2j; 1 ≤ j ≤ n.


ef (1) =

⌊4n+ 1

2

⌋, ef (0) =

⌈4n+ 1

2

⌉.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, S(K1,n)�K1 is SDC graph.

147


Example 6.2.9. SDC labeling in the graph S(K1,5) � K1,5 is demonstrated in the


u1

v0 v1 v1'

v5

v5'

v4

v4'

v3

v3'v2

v2'

u5

u5'

u1'

u4

u3

u3'

u2

u4'

u2'

v0'

1

3 2

10

9

8

6

7

4

5

12

20

13

2111

18

19

16

17

14

22

15

Figure 6.20: SDC labeling in S(K1,5)�K1,5

Theorem 6.2.10. G�K1 is SDC graph, where G is cycle with one chord.

Proof. Let V (G�K1) = {vj, v′j | 1 ≤ j ≤ n}, where vj(1 ≤ j ≤ n) are vertices of Cn

and v′j(1 ≤ j ≤ n) are the lately inserted vertices to construct the graph G�K1.

Let E(G �K1) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{vjv′j | 1 ≤ j ≤ n}⋃{v2vn},

where v2vn is the chord of Cn.

Also it is to be noted that, |V (G�K1)| = 2n and |E(G�K1)| = 2n+ 1.

Consider a bijection f : V (G�K1)→ {1, 2, 3, . . . , 2n} defined as below.

f(vj) = 2j − 1; 1 ≤ j ≤ n.

f(v′j) = 2j; 1 ≤ j ≤ n.


ef (1) =

⌈2n+ 1

2

⌉, ef (0) =

⌊2n+ 1

2

⌋.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, G�K1 is SDC graph, where G is cycle with one chord.

Example 6.2.10. SDC labeling in corona of C6 with one chord and K1 is demon-


148


v1

v1'

v2

v2'

v3

v3' v4

v4'

v5

v5'

v6

v6'1

2

3

4

5

67

8

10

9

1112

Figure 6.21: SDC labeling in corona of C6 with one chord and K1

Theorem 6.2.11. Cn,3 �K1 is SDC graph.

Proof. Let V (Cn,3 �K1) = {vj, v′j | 1 ≤ j ≤ n}, where vj(1 ≤ j ≤ n) are vertices of

Cn and v′j(1 ≤ j ≤ n) are lately inserted vertices to construct the graph Cn,3 �K1.

Let E(Cn,3 � K1) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{vjv′j | 1 ≤ j ≤

n}⋃{v2vn, v2vn−1}, where v2vn and v2vn−1 are chords of Cn.

Also it is to be noted that, |V (Cn,3 �K1)| = 2n and |E(Cn,3 �K1)| = 2n+ 2.

Consider a bijection f : V (Cn,3 �K1)→ {1, 2, 3, . . . , 2n} define defined as below.

Case 1: n ≡ 0, 1, 3(mod 4).

f(vj) =

j j ≡ 0, 1(mod 4);

j + 1 j ≡ 2(mod 4);

j − 1 j ≡ 3(mod 4); 1 ≤ j ≤ n.

f(v′j) = n+ j; 1 ≤ j ≤ n.

Case 2: n ≡ 2(mod 4).

f(vj) =

j j ≡ 0, 1(mod 4);

j + 1 j ≡ 2(mod 4);

j − 1 j ≡ 3(mod 4); 1 ≤ j ≤ n.

f(v′1) = n.

f(v′2) = n+ 2.

f(v′j) = n+ j; 3 ≤ j ≤ n.

149



ef (1) = n+ 1 = ef (0)

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Cn,3 �K1 is SDC graph.

Example 6.2.11. SDC labeling in the graph C7,3 �K1 is demonstrated in the fol-

lowing Figure 6.22.

v1

v1'

v2

v2'

v3v3'

v4

v4' v5

v5'

v6 v6'

v7

v7'1

8

3

9

210

4

11

5

12

7 13

6

14

Figure 6.22: SDC labeling in C7,3 �K1

Theorem 6.2.12. Cn(1, 1, n− 5)�K1 is SDC graph.

Proof. Let V (Cn(1, 1, n− 5)�K1) = {vj, v′j | 1 ≤ j ≤ n}, where vj(1 ≤ j ≤ n) are

the vertices of Cn and v′j(1 ≤ j ≤ n) are lately inserted vertices to construct the

graph Cn(1, 1, n− 5)�K1.

Let E(Cn(1, 1, n − 5) �K1) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{vjv′j | 1 ≤ j ≤

n}⋃{v1v3}⋃{v3vn−1}

⋃{vn−1v1}, where u1u3, u3un−1 and u1un−1 are chords of Cn

which by themselves form a triangle.

Also it is to be noted that, |V (Cn(1, 1, n− 5)�K1)| = 2n and |E(Cn(1, 1, n− 5)�K1)| = 2n+ 3.

Consider a bijection f : V (Cn(1, 1, n−5)�K1)→ {1, 2, 3, . . . , 2n} defined as below.

Case 1: n ≡ 0, 1, 3(mod 4).

f(v′j) = n+ j; 1 ≤ j ≤ n.s

150


f(vj) =

j j ≡ 0, 1(mod 4);

j + 1 j ≡ 2(mod 4);

j − 1 j ≡ 3(mod 4); 1 ≤ j ≤ n.

Case 2: n ≡ 2(mod 4).

f(v′1) = n.

f(v′2) = n+ 2.

f(vj) =

j i ≡ 0, 1(mod 4);

j + 1 i ≡ 2(mod 4);

j − 1 i ≡ 3(mod 4); 1 ≤ j ≤ n.

f(v′j) = n+ j; 3 ≤ j ≤ n.


ef (1) =

⌊2n+ 3

2

⌋, ef (0) =

⌈2n+ 3

2

⌉.

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, Cn(1, 1, n− 5)�K1 is SDC graph.

Example 6.2.12. SDC labeling in the graph C8(1, 1, 3)�K1 is demonstrated in the


v1

v1'

v2

v2'

v3v3'

v4

v4'

v5

v5'

v6

v6'

v8

v7'v7

v8'1

3

2

45

7

6

8

9

10

11

12

13

16

15

14

Figure 6.23: SDC labeling in C8(1, 1, 3)�K1

151

6.3. SDC Labeling With the Use of Switching of a Vertex in Cycle Allied Graphs

6.3 SDC Labeling With the Use of Switching of a Vertex in

Cycle Allied Graphs

Vaidya and Shah[49] derived some captivating results on divisor cordial labeling

of the graphs constructed from switching a vertex in different graphs. In current

segment we demonstrate some graphs constructed from switching invariance in cycle

allied graphs for SDC labeling.

Theorem 6.3.1. Gv is SDC, where G is cycle Cn with one chord.


Let V (G) = {vj | 1 ≤ j ≤ n}, where vj(1 ≤ j ≤ n) are vertices of Cn.

Let E(G) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{v2vn}, where v2vn is the chord of

Cn. WLOG let the switched vertex be v1 (of degree 2 or 3).

Let Gv1 denote the graph constructed from switching of vertex v1.

Corresponding to the vertices of different degree in Cn with one chord, it is required

to discuss following two cases. Case 1: d(v1) = 2.


E(Gv1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v2vn}⋃{v1vj | 3 ≤ j ≤ n− 1}.

It is to be noted that, |V (Gv1)| = n and |E(Gv1)| = 2n− 4.

Consider a bijection f from V (Gv1) to {1, 2, 3, . . . , n} defined as below.

Subcase 1: n ≡ 0(mod 4).

f(v1) = 1.

f(v2) = 2.

f(vj) =

j ; j ≡ 3(mod 4)

j + 1 ; j ≡ 1, 0(mod 4)

j + 2 ; j ≡ 2(mod 4); 3 ≤ j ≤ n− 1.

f(vn) = 4.

152



f(vj) =

j ; j ≡ 1, 2(mod 4)

j + 1 ; j ≡ 3(mod 4)

j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n.


f(v1) = n.

f(v2) = 2.

f(vj) =

j + 1 ; j ≡ 3(mod 4)

j − 1 ; j ≡ 0(mod 4)

j ; j ≡ 1, 2(mod 4); 3 ≤ j ≤ n− 1.

f(vn) = 1.


f(v1) = 1.

f(v2) = 2.

f(vj) =

j ; j ≡ 3(mod 4)

j + 1 ; j ≡ 0, 1(mod 4)

j + 2 ; j ≡ 2(mod 4); 3 ≤ j ≤ n− 2.

f(vn−1) = n.

f(vn) = 4.


ef (1) = ef (0) = n− 2.

Case 2: d(v1) = 3.


E(Gv1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v1vj | 4 ≤ j ≤ n− 1}.

153


Also it is to be noted that, |V (Gv1)| = n and |E(Gv1)| = 2n− 6.


Subcase 1: n ≡ 0, 1, 2(mod 4).

f(vj) =

j ; j ≡ 1, 2(mod 4)

j + 1 ; j ≡ 3(mod 4)

j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n.


f(vj) =

j ; j ≡ 1, 2(mod 4)

j + 1 ; j ≡ 3(mod 4)

j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n− 1.

f(vn) = n.

By looking into the above prescribed pattern, ef (1) = ef (0) = n− 3.

Then we get, |ef (0)− ef (1)| ≤ 1 in each case.

That is, Gv is SDC, where G is cycle Cn with one chord.


(i) Cycle C7 with one chord.

(ii) SDC labeling in (G)v, where d(v) = 2 and G is C7 with one chord.

(iii) SDC labeling in (G)v, where d(v) = 3 and G is C7 with one chord.

1

2

6

7

4

3

5

v6

v5v4

v3

v2

v1

v7

1

3

4

7

2

5 6

Figure 6.24: SDC labeling in (G)v.

154


Theorem 6.3.2. (Cn,3)v is SDC.

Proof. Let V (Cn,3) = {v1, v2, . . . , vn}, where vj(1 ≤ j ≤ n) are the vertices of Cn.

Let E(Cn,3) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{v2vn}

⋃{v2vn−1}, where v2vn,

v2vn−1 are chords.


Let (Cn,3)v1 denote the graph constructed from switching of vertex v1 of Cn,3.

Corresponding to the vertices of different degree in Cn,3, it is required to discuss


Case 1: d(v1) = 2.


E((Cn,3)v1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v2vn}⋃{v2vn−1}

⋃{v1vj | 3 ≤ j ≤ n− 1}.In this case it is to be noted that, |V ((Cn,3)v1)| = n and |E((Cn,3)v1)| = 2n− 3.

Consider a bijection f from V ((Cn,3)v1) to {1, 2, 3, . . . , n} defined as below.

Subcase 1: n ≡ 0, 1, 2(mod 4).

f(vj) =

j ; j ≡ 1, 2(mod 4)

j + 1 ; j ≡ 3(mod 4)

j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n.


f(vj) =

j ; j ≡ 1, 2(mod 4)

j + 1 ; j ≡ 3(mod 4)

j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n− 1.

f(vn) = n.



n ≡ 0, 1, 3(mod 4) ef (1) = n− 2, ef (0) = n− 1

n ≡ 2(mod 4) ef (1) = n− 1, ef (0) = n− 2

Case 2: d(v1) = 3.

155



E((Cn,3)v1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v3vn}⋃{v1vj | 4 ≤ j ≤ n− 1}.

In this case it is to be noted that, |V ((Cn,3)v1)| = n and |E((Cn,3)v1)| = 2n− 5.

Consider a bijection f from V ((Cn,3)v1) to {1, 2, 3, . . . , n} defined as below.

Subcase 1: n ≡ 0, 1, 2(mod 4).

f(vj) =

j ; j ≡ 1, 2(mod 4)

j + 1 ; j ≡ 3(mod 4)

j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n.


f(vj) =

j ; j ≡ 1, 2(mod 4)

j + 1 ; j ≡ 3(mod 4)

j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n− 1.

f(vn) = n.



n ≡ 0, 2, 3(mod 4) ef (1) = b2n−52c, ef (0) = d2n−5

2e

n ≡ 1(mod 4) ef (0) = b2n−52c, ef (1) = d2n−5

2e

Case 3: d(v1) = 4.


E((Cn,3)v1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v1vj | 3 ≤ j ≤ n− 3}.In this case it is to be noted that, |V ((Cn,3)v1)| = n and |E((Cn,3)v1)| = 2n− 7.

Consider a bijection f from V ((Cn,3)v1) to {1, 2, 3, . . . , n} define defined as below.

Subcase 1: n ≡ 0, 1, 2(mod 4).

f(vj) =

j ; j ≡ 1, 2(mod 4)

j + 1 ; j ≡ 3(mod 4)

j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n.

156



f(vn) = n.

f(vj) =

j ; j ≡ 1, 2(mod 4)

j + 1 ; j ≡ 3(mod 4)

j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n− 1.



n ≡ 0, 1(mod 4) ef (0) =⌊2n−7

2

⌋, ef (1) =

⌈2n−7

2

⌉

n ≡ 2, 3(mod 4) ef (1) =⌊2n−7

2

⌋, ef (0) =

⌈2n−7

2

⌉


That is, (Cn,3)v is SDC graph.


(i) The graph C8,3.

(ii) SDC labeling in (C8,3)v, where d(v) = 2.

(iii) SDC labeling in (C8,3)v, where d(v) = 3.

(iv) SDC labeling in (C8,3)v, where d(v) = 4.

1

5

72

4

3 6

8

1

5

7

2

4

3

6

8

v7

v6

v5

v4

v3

v1

v8v2 1

3

7

2

4 5

6

8

Figure 6.25: SDC labeling in (C8,3)v.

Theorem 6.3.3. (Cn(1, 1, n− 5))v is SDC (n ≥ 6, n ∈ N) .

Proof. Let Cn(1, 1, n− 5) be denoted as G.

Let V (G) = {vj | 1 ≤ j ≤ n} = V (Cn).

157


Let E(G) = {vjvj+1 | 1 ≤ j ≤ n−1}⋃{vnv1}⋃{v1v3}

⋃{v3vn−1}⋃{vn−1v1}, where

v1vn−1, v1v3, vn−1v3 are chords.


Let Gv1 denote the graph constructed from switching of arbitrary vertex v1 of G.

As per different possible degrees of vertices in the graph G, we need to consider


Corresponding to the vertices of different degree in Cn(1, 1, n− 5), it is required to

discuss following two cases.

Case 1: d(v1) = 2.


E(Gv1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v2v4}⋃{v4vn−1}

⋃{vn−1v2}⋃{v1vj | 3 ≤ j ≤

n− 1}.In this case it is to be noted that, |V (Gv1)| = n and |E(Gv1)| = 2n− 2.


Subcase 1: n ≡ 1, 2(mod 4).

f(vj) =

j ; j ≡ 1, 2(mod 4)

j + 1 ; j ≡ 3(mod 4)

j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n.


f(v1) = n− 1.

f(vj) =

j − 1 ; j ≡ 1, 2(mod 4)

j ; j ≡ 3(mod 4)

j − 2 ; j ≡ 0(mod 4); 2 ≤ j ≤ n.


Case 2: d(v1) = 4.


E(Gv1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v3vn−1}⋃{v1vj | 4 ≤ j ≤ n− 2}.

In this case it is to be noted that, |V (Gv1)| = n and |E(Gv1)| = 2n− 6.

158




f(vj) =

j ; j ≡ 1, 2(mod 4)

j + 1 ; j ≡ 3(mod 4)

j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n.


f(v1) = n− 1.

f(vj) =

j ; j ≡ 1, 2(mod 4)

j + 1 ; j ≡ 3(mod 4)

j − 1 ; j ≡ 0(mod 4); 2 ≤ j ≤ n− 2.

f(vn−1) = n.

f(vn) = 1.



That is, (Cn(1, 1, n− 5))v is SDC.


(i) The graph C7(1, 1, 2).

(ii) SDC labeling in (C7(1, 1, 2))v, where d(v) = 2.

(iii) SDC labeling in (C7(1, 1, 2))v, where d(v) = 4.

v6

v5v4

v3

v2

v1

v7 1

7

6

53

4

2

1

4

7

6

5

2

3

Figure 6.26: SDC labeling in (C7(1, 1, 2))v

159

6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs

6.4 SDC Labeling With the Use of Switching of a Vertex in

Wheel and Shell Allied Graphs

In the current segment we demonstrate some new graphs constructed from switching

of a vertex in wheel and shell allied graphs for SDC labeling.

Theorem 6.4.1. (Wn)v is SDC, where v is rim vertex.

Proof. Let V (Wn) = {vj | 0 ≤ j ≤ n}, where v0 is apex vertex and vj(1 ≤ j ≤ n)

are the rim vertices of wheel Wn.


Let (Wn)v1 denote the graph constructed from switching of a rim vertex v1 of Wn.

Then by the effect of switching operation, the edge set of (Wn)v1 is

E((Wn)v1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v0vj | 2 ≤ j ≤ n}⋃{v1vj | 3 ≤ j ≤ n− 1}.In this case it is to be noted that, |V ((Wn)v1)| = n+ 1 and |E((Wn)v1)| = 3n− 6.

Consider a bijection f from V ((Wn)v1) to {1, 2, 3, . . . , n+ 1} defined as below.

Case 1: n ≡ 0, 1(mod 4).

f(v0) = 1.

f(vj) =

j + 1 ; j ≡ 1, 2(mod 4)

j + 2 ; j ≡ 3(mod 4)

j ; j ≡ 0(mod 4); 1 ≤ j ≤ n.

Case 2: n ≡ 2(mod 4).

f(v0) = 1.

f(v1) = 3.

f(vj) =

j ; j ≡ 2(mod 4)

j + 1 ; j ≡ 0, 3(mod 4)

j + 2 ; j ≡ 1(mod 4); 2 ≤ j ≤ n.

160


Case 3: n ≡ 3(mod 4).

f(v0) = 1.

f(vj) =

j + 1 ; j ≡ 1, 2(mod 4)

j + 2 ; j ≡ 3(mod 4)

j ; j ≡ 0(mod 4); 1 ≤ j ≤ n− 1.

f(vn) = n+ 1.



n ≡ 0, 2(mod 4) ef (0) = 3n−62

= ef (1)

n ≡ 1(mod 4) ef (0) =⌊3n−6

2

⌋, ef (1) =

⌈3n−6

2

⌉

n ≡ 3(mod 4) ef (0) =⌈3n−6

2

⌉, ef (1) =

⌊3n−6

2

⌋

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, (Wn)v is SDC, where v is rim vertex.


(i) Wheel W9.

(ii) SDC labeling in (W9)v1, where v1 is rim vertex.

v1

v9

v8

v7

v6v5

v4

v3

v2

v0

9

7

5

3

1

2

4

6

8

10

Figure 6.27: SDC labeling in (W9)v1 .

Remark 6.4.1. Switching apex vertex in Wn, the resultant graph is Cn⋃K1 which

is SDC graph!!

161


Theorem 6.4.2. (Gn)v is SDC, where v is not apex vertex.

Proof. Let V (Gn) = {vj | 0 ≤ j ≤ 2n}, where v0 is apex vertex and vj(1 ≤ j ≤ 2n)

are other vertices of Gn, where

deg(vj) =

2 when j is even;

3 when j is odd; 1 ≤ j ≤ 2n.

Let E(Gn) = {vjvj+1 | 1 ≤ j ≤ 2n− 1}⋃{v2nv1}⋃{v0v2j−1 | 1 ≤ j ≤ n}.

(Gn)vi∼= (Gn)vj , where d(vi) = d(vj).

Let (Gn)vj denote the graph constructed from switching of vertex vj (j = 1, 2) of

Gn.



Case 1: deg(v1) = 3.

Then by the effect of switching operation, the edge set of (Gn)v1 is

E((Gn)v1) = {v0v2j−1 | 2 ≤ j ≤ n}⋃{vjvj+1 | 2 ≤ j ≤ 2n − 1}⋃{v1vj | 3 ≤ j ≤2n− 1}.In this case it is to be noted that, |V (Gn)v1)| = 2n+ 1 and |E(Gn))v1)| = 5n− 6.

Consider a bijection f from V ((Gn))v1)) to {1, 2, . . . , 2n+ 1} defined as below.

f(v0) = 1.

f(vj) =

j + 1 ; j ≡ 1, 2(mod 4)

j + 2 ; j ≡ 3(mod 4)

j ; j ≡ 0(mod 4); 1 ≤ j ≤ n.



n ≡ 0, 2(mod 4) ef (1) = 5n−62

= ef (0)

n ≡ 1, 3(mod 4) ef (0) =⌊5n−6

2

⌋, ef (1) =

⌈5n−6

2

⌉


Then by the effect of switching operation, the edge set of (Gn)v2 is

E((Gn)v2) = {v0v2j−1 | 1 ≤ j ≤ n}⋃{vjvj+1 | 3 ≤ j ≤ 2n − 1}⋃{v2nv1}⋃{v2vj |

162


4 ≤ j ≤ 2n}.In this case it is to be noted that, |V ((Gn)v2)| = 2n+ 1 and |E((Gn)v2)| = 5n− 4.

Consider a bijection f from V ((Gn)v2) to {1, 2, . . . , 2n+ 1} defined as below.


f(v0) = 1.

f(vj) =

j + 1 ; j ≡ 1, 2(mod 4)

j + 2 ; j ≡ 3(mod 4)

j ; j ≡ 0(mod 4); 1 ≤ j ≤ n.


f(v0) = 1.

f(v1) = 3.

f(vj) =

j + 2 ; j ≡ 3(mod 4)

j ; j ≡ 2(mod 4)

j + 1 ; j ≡ 0, 1(mod 4); 2 ≤ j ≤ n.



n ≡ 0, 2(mod 4) ef (1) =⌊5n−4

2

⌋, ef (0) =

⌈5n−4

2

⌉

n ≡ 1, 3(mod 4) ef (1) = 5n−42

= ef (0)


That is, (Gn)v is SDC, v is not apex vertex.


(i) Gear graph G6.

(ii) SDC labeling in (G6)v, where d(v) = 3.

(iii) SDC labeling in (G6)v, where d(v) = 2.

163


v11

v0

v12

v10

v9

v8

v7

v6

v5

v4

v2

v1

v3 13

7

1

5

9

2

3

4

68

10

11

12 2

1

3

5

46

79

8

10

11

1312

Figure 6.28: SDC labeling in (G6)v.

Remark 6.4.2. Switching apex vertex in Gn, the resultant graph is C2n

⋃K1 which

is SDC graph!!

Theorem 6.4.3. (Sn)v is SDC, where v is not apex vertex.

Proof. Let V (Sn) = {vi | 1 ≤ i ≤ n}, where v1 is apex vertex and vj(2 ≤ j ≤ n) are

the other vertices of shell Sn, where

deg(vj) =

2 when j = 2, n.

3 when ; 3 ≤ j ≤ n− 1.

Let E(Sn) = {vjvj+1 | 1 ≤ j ≤ n− 1}⋃{vnv1}⋃{v1vj | 3 ≤ j ≤ n− 1}.

(Sn)vi∼= (Sn)vj , where d(vi) = d(vj).

Let (Sn)vj denote the graph constructed from switching of vertex vj (j = 2, 3) of Sn.

Corresponding to the vertices of different degree in Sn, it is required to discuss



Then by the effect of switching operation, the edge set of (Sn)v3 is

Let E((Sn)v3) = {v1vj | 4 ≤ j ≤ n− 1}⋃{vjvj+1 | 4 ≤ j ≤ n− 1}⋃{vnv1}{v1v2}⋃

{v3vj | 5 ≤ j ≤ n}.In this case it is to be noted that, |V ((Sn)v3)| = n and |E((Sn)v3)| = 3n− 10.

Consider a bijection f from V ((Sn)v3) to {1, 2, . . . n} defined as below.


f(v1) = 1.

164


f(vn) = n− 1.

f(vn−1) = 2.

f(vj) =

j ; j ≡ 3(mod 4)

j + 1 ; j ≡ 0, 1(mod 4)

j + 2 ; j ≡ 2(mod 4); 2 ≤ j ≤ n− 2.


f(v1) = 1.

f(vn) = n.

f(vn−1) = 2.

f(vj) =

j ; j ≡ 3(mod 4)

j+ ; j ≡ 0, 1(mod 4)

j + 2 ; j ≡ 2(mod 4); 2 ≤ j ≤ n− 2.


f(v1) = 1.

f(vn) = n.

f(vn−1) = 2.

f(vj) =

j + 1 ; j ≡ 1, 2(mod 4)

j + 2 ; j ≡ 3(mod 4)

j ; j ≡ 0(mod 4); 2 ≤ j ≤ n− 2.



n ≡ 0, 2(mod 4) ef (1) = 3n−102

= ef (0)

n ≡ 1, 3(mod 4) ef (1) =⌊3n−10

2

⌋, ef (0) =

⌈3n−10

2

⌉


Then by the effect of switching operation, the edge set of (Sn)v2 is

165


Let E((Sn)v2) = {v1vj | 3 ≤ j ≤ n − 1}⋃{vjvj+1 | 3 ≤ j ≤ n − 1}⋃{vnv1}{v2vj |4 ≤ j ≤ n}.In this case it is to be noted that, |V ((Sn)v2)| = n, |E((Sn)v2)| = 3n− 8.

Consider a bijection f from V ((Sn)v2) to {1, 2, . . . , n} defined as below.

Subcase 1: n ≡ 0, 1, 3(mod 4).

f(v1) = 1.

f(vn) = 2.

f(vj) =

j + 2 ; j + 2 ≡ 2(mod 4)

j ; j ≡ 3(mod 4)

j + 1 ; j ≡ 0, 1(mod 4); 3 ≤ j ≤ n− 1.


f(v1) = 1.

f(v2) = 2.

f(vj) =

j + 1 ; j ≡ 1, 2(mod 4)

j + 2 ; j ≡ 3(mod 4)

j ; j ≡ 0(mod 4); 3 ≤ j ≤ n− 1.

f(vn) = n.



n ≡ 0(mod 4) ef (0) =⌊3n−8

2

⌋, ef (1) =

⌈3n−8

2

⌉

n ≡ 1, 3(mod 4) ef (1) = 3n−82

= ef (0)

n ≡ 2(mod 4) ef (1) =⌊3n−8

2

⌋, ef (0) =

⌈3n−8

2

⌉


That is, (Sn)v is SDC, v is not apex vertex.


(i) Shell graph S7.

166


(ii) SDC labeling in (S7)v, where d(v) = 3.

(iii) SDC labeling in (S7)v, where d(v) = 2.

v6

v5v4

v3

v7

v1

v2

2

1

3

4

5

6

72

1

4 3

5

67

Figure 6.29: SDC labeling in (S7)v

Remark 6.4.3. Switching apex vertex in Sn, the resultant graph is Pn−2⋃K1 which

is SDC graph!!

Theorem 6.4.4. (Hn)v is SDC, v is apex vertex.

Proof. Let V (Hn) = {v0, vj, uj | 1 ≤ j ≤ n}, where v0 is apex vertex, vj(1 ≤ j ≤ n)

are the vertices of corresponding to cycle Cn and uj(1 ≤ j ≤ n) are the pendant

vertices.

E(Hn) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{v0vj | 1 ≤ j ≤ n}⋃{vjuj | 1 ≤ j ≤

n}.Let (Hn)v0 denote the graph constructed from switching of apex vertex v0 of Hn.

Then by the effect of switching operation, the edge set of (Hn)v0 is

Let E((Hn)v0) = {v0uj | 1 ≤ j ≤ n}⋃{vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}{vjuj | 1 ≤j ≤ n}.In this case it is to be noted that, |V ((Hn)v0)| = 2n+ 1, |E((Hn)v0)| = 3n.

Consider a bijection f from V ((Hn)v0) to {1, 2, . . . 2n+ 1} defined as below.

Case 1: n ≡ 0, 2(mod 4).

f(v0) = 2n+ 1.

f(vj) = j; 1 ≤ j ≤ n.

f(uk) = n+ 1 + k; 1 ≤ k ≤ n.

167


Case 2: n ≡ 1, 3(mod 4).

f(v0) = 2n.

f(vj) =

2j − 1 ; j ≡ 1, 3(mod 4)

2j − 2 ; j ≡ 0, 2(mod 4); 1 ≤ j ≤ n.

f(uk) =

2k + 1 ; k ≡ 1, 3(mod 4)

2k ; k ≡ 0, 2(mod 4); 1 ≤ k ≤ n.



n ≡ 0, 2(mod 4) ef (1) = 3n2

= ef (0)

n ≡ 1, 3(mod 4) ef (0) =⌊3n2

⌋, ef (1) =

⌈3n2

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, (Hn)v is SDC, where v is apex vertex.


(i) Helm graph H6.

(ii) SDC labeling in (H6)v, where v is apex vertex.

13v0

v5

v4

v3

v1

v6

u5

u4

u3

u2

u1

u6

v2

1

2

3

4

6

7

8

9

10

12

5

11

Figure 6.30: SDC labeling in (H6)v.

Theorem 6.4.5. (CHn)v is SDC, where v is apex vertex.

Proof. Let V (CHn) = {v0, vj, uj | 1 ≤ j ≤ n}, where v0 is apex vertex, vj(1 ≤ j ≤ n)

are the vertices of inner cycle and uj(1 ≤ j ≤ n) are the vertices of outer cycle of

168


CHn.

Let E(CHn) = {vjvj+1, vnv1 | 1 ≤ j ≤ n−1}⋃{v0vj, vjuj | 1 ≤ j ≤ n}⋃{ujuj+1, unu1 |1 ≤ j ≤ n− 1}.Let (CHn)v0 denote the graph constructed from switching of apex vertex v0.

Then by the effect of switching operation, the edge set of (CHn)v0 is

Let E((CHn)v0) = {v0uj | 1 ≤ j ≤ n}⋃{vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}{vjuj |1 ≤ j ≤ n}{ujuj+1 | 1 ≤ j ≤ n− 1}⋃{unu1}.In this case it is to be noted that, |V ((CHn)v0)| = 2n+ 1 and |E((CHn)v0)| = 4n.

Consider a bijection f from V ((CHn)v0) to {1, 2, . . . 2n+ 1} define defined as below.

f(v0) = 1.

f(vj) = 2j + 1; 1 ≤ j ≤ n.

f(uk) = 2k; 1 ≤ k ≤ n.


ef (1) = 2n = ef (1).

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, (CHn)v is SDC, where v is apex vertex.


(i) Closed helm graph CH6.

(ii) SDC labeling in (CH6)v, where v is apex vertex.

v0

12

13

10

11

8

9 6

7

5

4

2

v5

v4

v3

v2

v1

v6

u5

u4

u3

u2

u1

u6

1

3

Figure 6.31: SDC labeling in (CH6)v

169


Theorem 6.4.6. (Fln)v is SDC, where v is not apex vertex.

Proof. Let V (Fln) = {u0}⋃{uj | 1 ≤ j ≤ n}⋃{vj | 1 ≤ j ≤ n}, where u0 is apex

vertex, uj(1 ≤ j ≤ n) are the internal vertices and vj(1 ≤ j ≤ n) are the external

vertices. deg(uj) = 4; deg(vj) = 2, 1 ≤ j ≤ n.

Let E(Fln) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{unu1}⋃{u0uj | 1 ≤ j ≤ n}⋃{u0vj | 1 ≤

j ≤ n}⋃{ujvj | 1 ≤ j ≤ n}(Fln)ui

∼= (Fln)uj , where d(ui) = d(uj).







E((Fln)v1) = {ujuj+1 | 1 ≤ j ≤ n}⋃{u0uj | 1 ≤ j ≤ n}⋃{u0vj | 2 ≤ j ≤n}⋃{ujvj | 2 ≤ j ≤ n}⋃{v1uj | 2 ≤ j ≤ n− 1}⋃{v1vj | 2 ≤ j ≤ n}.In this case it is to be noted that, |V ((Fln)v1)| = 2n+ 1 and |E((Fln)v1)| = 6n− 4.

Consider a bijection f from V ((Fln)v1) to {1, 2, . . . 2n+ 1} defined as below.

f(u0) = 2.

f(v1) = 1.

f(uj) = 2j + 1; 1 ≤ j ≤ n.

f(vj) = 2j; 2 ≤ j ≤ n.


ef (1) = 3n− 2 = ef (0).


Then by the effect of switching operation, the edge set of (Fln)u1 is

E((Fln)u1) = {ujuj+1, unu1 | 2 ≤ j ≤ n− 1}⋃{u0uj | 2 ≤ j ≤ n}⋃{u0vj | 1 ≤ j ≤n}⋃{ujvj | 2 ≤ j ≤ n}⋃{u1uj | 3 ≤ j ≤ n− 1}⋃{u1vj | 2 ≤ j ≤ n}.In this case it is to be noted that, |V ((Fln)u1)| = 2n+ 1, |E((Fln)u1)| = 6n− 8.

170


Consider a bijection f from V ((Fln)u1) to {1, 2, . . . 2n+ 1} defined as below.

f(u0) = 2.

f(u1) = 1.

f(v1) = 2n+ 1.

f(uj) = 2j; 2 ≤ j ≤ n.

f(vj) = 2j − 1; 2 ≤ j ≤ n.


ef (1) = 3n− 4 = ef (0).


That is, (Fln)v is SDC, v is not apex vertex.


(i) Flower graph Fl4.

(ii) SDC labeling in (Fl4)v1, where d(v1) = 2.

(iii) SDC labeling in (Fl4)u1, where d(u1) = 4.

1

3

8

2

9`

64

5 7

2

1

7

8

5`

64

9

3

v1v4

u1 u4

v2v3

u2 u3

u0

Figure 6.32: SDC labeling in (Fl4)v

Remark 6.4.4. Switching apex vertex in Fln, the resultant graph is (Cn�K1)⋃K1

which is SDC graph!!

171


Theorem 6.4.7. (Bm,n)v is SDC.

Proof. Let V (Bm,n) = {u0, v0, ui, vj | 1 ≤ i ≤ m, 1 ≤ j ≤ n}, where u0, v0 are apex

vertices and d(ui) = 1(1 ≤ i ≤ m) and d(vj) = 1(1 ≤ j ≤ n).

E(Bm,n) = {u0ui, v0vj, u0v0 | 1 ≤ i ≤ m, 1 ≤ j ≤ n}.WLOG, let us assume that m ≤ n (as Bm,n and Bn,m are isomorphic graphs).

Let (Bm,n)v denote the graph constructed from switching of an arbitrary vertex v

in Bm,n.

According to different degrees of vertices in (Bm,n)v it is required to discuss following

three cases.

Case 1: deg(vj) = 1.

WLOG, let us assume that v1 is the switched pendant vertex.

In this case it is to be noted that, |V ((Bm,n)v1)| = m+n+2, |E((Bm,n)v1)| = 2m+2n.

Consider a bijection f from V ((Bm,n)v1) to {1, 2, . . .m+ n+ 2} defined as below.

f(u0) = 2.

f(u1) = 3.

f(v0) = 4.

f(v1) = 1.

f(ui) = 3 + i; 2 ≤ i ≤ m.

f(vj) = 2 +m+ j; 2 ≤ j ≤ n.


ef (1) = m+ n = ef (0).

Case 2: deg(u0) = m.

In this case it is to be noted that, |V ((Bm,n)u0)| = m+n+2 and |E((Bm,n)u0)| = 2n.

Consider a bijection f from V ((Bm,n)u0) to {1, 2, . . . ,m+ n+ 2} defined as below.

f(u0) = 2.

f(v0) = 1.

f(ui) = 2 + n+ i; 1 ≤ i ≤ m.

172


f(vj) = 2 + j; 1 ≤ j ≤ n.


ef (1) = n = ef (0).

Then we get, |ef (0)− ef (1)| ≤ 1.

Case 3: Switching of apex vertex v0 of degree n.

In this case it is to be noted that, |V ((Bm,n)v0)| = m+n+2 and |E((Bm,n)v0)| = 2m.

Consider a bijection f from V ((Bm,n)v0) to {1, 2, . . .m+ n+ 2} defined as below.

f(u0) = 2.

f(v0) = 1.

f(ui) = 2 + n+ i; 1 ≤ i ≤ m.

f(vj) = 2 + j; 1 ≤ j ≤ n.

By looking into the above prescribed pattern, ef (1) = m = ef (0).

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, (Bm,n)v is SDC.

Example 6.4.7. Bistar B4,5 and SDC labeling in (B4,5)v1, where d(v1) = 1 are

demonstrated in the following Figure 6.33.

u0

v0

u2

u3

v1

u4

v2

v3

v4

v5

u1

2

4

3

5

6

7

1

8

9

10

11

Figure 6.33: SDC labeling in (B4,5)v1

Example 6.4.8. Bistar B4,5 and SDC labeling in (B4,5)u0, where u0 is apex vertex

are demonstrated in the following Figure 6.34.

173


u1

u2

v1

v2

u3

u4

v3

v4

v5

v0

u0

1

2

3

4

5

6

7

8

9

10

11

Figure 6.34: SDC labeling in (B4,5)u0

Example 6.4.9. Bistar B4,5 and SDC labeling in (B4,5)v0, where v0 is apex vertex


u1

u2

v1

v2

u3

u4

v3

v4

v5

v0

u0

1

2

3

4

5

6

7

8

9

10

11

Figure 6.35: SDC labeling in (B4,5)v0

Theorem 6.4.8. (Pn �K1)v is SDC.

Proof. Let V (Pn�K1) = {uj, vj : 1 ≤ j ≤ n}, where vj are pendant vertices and uj

are vertices of path Pn ;j = 1, 2, . . . , n.

E(Pn �K1) = {ujuj+1 : 1 ≤ j ≤ n− 1}⋃{ujvj : 1 ≤ j ≤ n}.Let (Pn �K1)v denote the graph constructed from switching of arbitrary vertex v

in Pn �K1.

According to different degrees of vertices of (Pn � K1)v, it is required to discuss



WLOG, let us assume that v1 is the switched pendant vertex.

174


In this case it is to be noted that, |V (Pn�K1)v1| = 2n and |E(Pn�K1)v1 | = 4n−4.

Consider a bijection f from V ((Pn �K1)v1) to {1, 2, . . . 2n} defined as below.

f(uj) = 2j − 1; 1 ≤ j ≤ n.

f(vk) = 2k; 1 ≤ k ≤ n.


ef (1) = 2n− 2 = ef (0).


WLOG, let us assume that the switched vertex is u1.

In this case it is to be noted that, |V (Pn�K1)u1)| = 2n and |E(Pn�K1)u1)| = 4n−6.

Consider a bijection f from V (Pn �K1)u1) to {1, 2, . . . 2n} defined as below.

f(u1) = 2.

f(v1) = 1.

f(uj) = 2j − 1; 2 ≤ j ≤ n.

f(vk) = 2k; 2 ≤ k ≤ n.


ef (1) = 2n− 3 = ef (0).



In this case it is to be noted that, |V (Pn�K1)u2)| = 2n and |E(Pn�K1)u2)| = 4n−8.

Consider a bijection f from V (Pn �K1)u2) to {1, 2, . . . 2n} defined as below.

f(u1) = 3.

f(u2) = 2.

f(v1) = 4.

f(v2) = 1.

175


f(uj) = 2j − 1; 3 ≤ j ≤ n.

f(vk) = 2k; 3 ≤ k ≤ n.


ef (1) = 2n− 4 = ef (1).


That is, (Pn �K1)v is SDC.

Example 6.4.10. Comb P5 �K1 and SDC labeling in (P5 �K1)v1, where (v1) = 1


u1 u2 u3 u4 u5

v1 v2 v3 v4 v5

1 3 5 9

2 4 6 8 10

7

Figure 6.36: SDC labeling in (P5 �K1)v1

Example 6.4.11. Comb P5 �K1 and SDC labeling in (P5 �K1)u1, where (u1) = 2


u1 u2 u3 u4 u5

v1 v2 v3 v4 v5 1

2 3 5 97

4 6 8 10

Figure 6.37: SDC labeling in (P5 �K1)u1

Example 6.4.12. Comb P5 �K1 and SDC labeling in (P5 �K1)u2, where (u2) = 3


176


u1 u2 u3 u4 u5

v1 v2 v3 v4 v5

2

1

973 5

4 6 8 10

Figure 6.38: SDC labeling in (P5 �K1)u2

Theorem 6.4.9. (Cn �K1)v is SDC.

Proof. Let V (Cn�K1) = {vj, uj | 1 ≤ j ≤ n}, where vj are pendant vertices and uj

are vertices of degree 3, j = 1, 2, . . . , n.

E(Cn �K1) = {ujuj+1, unu1 | 1 ≤ j ≤ n− 1},⋃{ujvj | 1 ≤ j ≤ n}.Let (Cn�K1)v be the graph constructed from switching of an arbitrary vertex v in

Cn �K1.

Corresponding to the vertices of different degree in Cn�K1, it is required to discuss



WLOG, let us assume that the switched pendant vertex is v1.

In this case it is to be noted that, |V (Cn�K1)v1| = 2n and |E(Cn�K1)v1 | = 4n−3.

Consider a bijection f from V ((Cn �K1)v1) to {1, 2, . . . 2n} defined as below.

f(vj) = 2j − 1; 1 ≤ j ≤ n.

f(uj) = 2j; 1 ≤ j ≤ n.


ef (1) = 2n− 1, ef (1) = 2n− 2.



In this case it is to be noted that, |V (Cn�K1)u1 | = 2n and |E(Cn�K1)u1| = 4n−7.

Consider a bijection f from V ((Cn �K1)u1) to {1, 2, . . . 2n} defined as below.

f(u1) = 1.

f(v1) = 2.

177


f(uj) = 2j; 2 ≤ j ≤ n.

f(vj) = 2j − 1; 2 ≤ j ≤ n.


ef (1) = 2n− 3, ef (0) = 2n− 4.


That is, (Cn �K1)v is SDC.

Example 6.4.13. Crown C7�K1 and SDC labeling in (C7�K1)v1, where (v1) = 1


v4

u1

v7

u7

v3

u6

u5u4

u3

v2

u2

v6

v5

v1

v1v7

v6

v5

v4

v3

v2

u1

u7

u6

u5u4

u3

u2

1

2

4

6

8 10

12

14

3

5

7

9

11

13

Figure 6.39: SDC labeling in (C7 �K1)v1

Example 6.4.14. Crown C7�K1 and SDC labeling in (C7�K1)u1, where (u1) = 3


13

v4

u1

v7

u7

v3

u6

u5u4

u3

v2

u2

v6

v5

v1

v7

v6

v5

v4

v3

v2

u7

u6

u5u4

u3

u2

1

2

4

6

8 10

12

145

7

9

11

3v1

u1

Figure 6.40: SDC labeling in (C7 �K1)u1

178


Theorem 6.4.10. (ACn)v is SDC, where ACn = Cn � P2.

Proof. Let V (ACn) = {vj, wj, uj | 1 ≤ j ≤ n}, where vj, wj and uj are vertices of

degree 1, 2 and 3 respectively; j = 1, 2, . . . , n.

E(ACn) = {ujuj+1, unu1 | 1 ≤ j ≤ n− 1}⋃{ujwj, wjvj | 1 ≤ j ≤ n}.Let (ACn)v denote the graph constructed from switching of an arbitrary vertex v in

ACn.

According to different degrees of vertices of the graph (ACn)v, it is required to

discuss following three cases.


WLOG, let us assume that the switched pendant vertex is v1.

In this case it is to be noted that, |V ((ACn)v1)| = 3n and |E((ACn)v1)| = 6n− 3.

Consider a bijection f from V ((ACn)v1) to {1, 2, . . . , 3n} defined as below.

f(v1) = 1.

f(vj) = 2n+ j; 2 ≤ j ≤ n.

f(uj) = 2j + 1; 1 ≤ j ≤ n.

f(wj) = 2j; 1 ≤ j ≤ n.


ef (1) = 3n− 1, ef (1) = 3n− 2.

Case 2: deg(w1) = 2.

WLOG, let us assume that the switched vertex is w1.

In this case it is to be noted that, |V ((ACn)w1)| = 3n and |E((ACn)w1)| = 6n− 5.

Consider a bijection f from V ((ACn)w1) to {1, 2, . . . 3n} defined as below.

f(v1) = 2n.

f(w1) = 1.

f(uj) = 2j + 1; 1 ≤ j ≤ n.

f(wj) = 2(j − 1); 2 ≤ j ≤ n.

f(vj) = 2n+ j; 2 ≤ j ≤ n.

179



ef (1) = 3n− 2, ef (0) = 3n− 3.



In this case it is to be noted that, |V ((ACn)u1)| = 3n and |E((ACn)u1)| = 6n− 7.

Consider a bijection f from V ((ACn)u1) to {1, 2, . . . 3n} defined as below.

For n ≤ 7 :

Subcase 1: n ≡ 0, 2(mod 4)

f(u1) = 1.

f(uj) = 2n+ j; 2 ≤ j ≤ n.

f(w2j−1) = 4j − 2; 1 ≤ j ≤ n

2.

f(w2j) = 4j − 1; 1 ≤ j ≤ n

2.

f(v2j−1) = 4j; 1 ≤ j ≤ n

2.

f(v2j) = 4j + 1; 1 ≤ j ≤ n

2.


f(u1) = 1.

f(u2) = 2n+ 1.

f(u3) = 2n+ 3.

f(uj) = 2n+ j; 4 ≤ j ≤ n.

f(w2j−1) = 4j − 2; 1 ≤ j ≤⌈n

2

⌉.

f(w2j) = 4j − 1; 1 ≤ j ≤⌊n

2

⌋.

f(v2j−1) = 4j; 1 ≤ j ≤⌈n

2

⌉.

f(v2j) = 4j + 1; 1 ≤ j ≤⌊n

2

⌋.

180



ef (1) =

⌊6n− 5

2

⌋, ef (0) =

⌈6n− 5

2

⌉.

For n > 7 :


f(u1) = 1.

f(u2) = 2n+ 3.

f(u3) = 2n+ 2.

f(uj) = 2n+ j; 4 ≤ j ≤ n.

f(w2j−1) = 4j − 2; 1 ≤ j ≤ n

2.

f(w2j) = 4j − 1; 1 ≤ j ≤ n

2.

f(v2j−1) = 4j; 1 ≤ j ≤ n

2.

f(v2j) = 4j + 1; 1 ≤ j ≤ n

2.


f(u1) = 1.

f(u2) = 2n+ 1.

f(u3) = 2n+ 4.

f(u4) = 2n+ 3.

f(uj) = 2n+ j; 4 ≤ j ≤ n.

f(w2j−1) = 4j − 2; 1 ≤ j ≤⌈n

2

⌉.

f(w2j) = 4j − 1; 1 ≤ j ≤⌊n

2

⌋.

f(v2j−1) = 4j; 1 ≤ j ≤⌈n

2

⌉.

f(v2j) = 4j + 1; 1 ≤ j ≤⌊n

2

⌋.

181



ef (0) =

⌊6n− 5

2

⌋, ef (1) =

⌈6n− 5

2

⌉.


That is, (ACn)v is SDC.

Example 6.4.15. Armed crown AC5 and SDC labeling in (AC5)v1, where deg(v1) =


u2

v1

v5

u1

u5

u4u3

v4

v3

v2

w1

w4

w2

w3

1

7 9

3

11

2

10

86

4 5

13

14

15

12

w5u2

u1

u5

u4

u3

v1

v5

v4

v3

v2

w1

w4

w2

w3

w5

Figure 6.41: SDC labeling in (AC5)v1

Example 6.4.16. Armed crown AC5 and SDC labeling in (AC5)w1, where deg(w1) =


1

7 9

3

11 58

64

2

10

13

14

15

12

w1

u2

u1

u5

u4

u3

w4

w2

w3

w5

v1

v5

v4

v3

v2

u2

v1

v5

u1

u5

u4u3

v4

v3

v2

w1

w4

w2

w3

w5

Figure 6.42: SDC labeling in (AC5)w1

Example 6.4.17. Armed crown AC5 and SDC labeling in (AC5)u1, where deg(u1) =


182

6.5. SDC Labeling by Duplicating a Vertex/Edge in Star Graph

u2

v1

v5

u1

u5

u4u3

v4

v3

v2

w1

w4

w2

w3

w5

1

8

6

10

12

15

24

79

5

13 14

3

11

w1

u2

u1

u5

u4

u3

w4

w2

w3

w5

v1

v5

v4

v3

v2

Figure 6.43: SDC labeling in (AC5)u1

6.5 SDC Labeling by Duplicating a Vertex/Edge in Star

Graph

Vaidya and Prajapati[55] derived some attractive results on prime labeling of graphs

constructed by duplicating the graph elements. In this segment we demonstrate

some SDC graphs constructed by duplicating vertex/edge in star K1,n.

Theorem 6.5.1. The graph constructed by duplicating any vertex in star K1,n is

SDC graph.

Proof. Let V (K1,n) = {vj | 0 ≤ j ≤ n}, where v0 is apex vertex and vj(1 ≤ j ≤ n)

are pendant vertices of K1,n and E(K1,n) = {v0vj | 1 ≤ j ≤ n}.Let G denote the graph constructed by duplicating any vertex vj by vertex v′j in

K1,n.



Case 1: deg(v0) = n (duplicating apex vertex).

The graph constructed by duplicating apex vertex v0 in K1,n is the graph K2,n and

hence it is SDC graph (Refer [44]).

Case 2: deg(vj) = 1 (duplicating pendant vertex).

The graph constructed by duplicating any pendant vertex in K1,n is a star graph

K1,n+1 and hence it is SDC graph (Refer [44]).

Theorem 6.5.2. The graph constructed by duplicating any edge in star K1,n is SDC.

183



are pendant vertices of K1,n.

E(K1,n) = {v0vi | 1 ≤ i ≤ n}.Let G denote the graph constructed by duplicating an edge, say v0vn by a new edge

v′0v′n in K1,n.

Hence in G, d(v0) = n, d(v′0) = n, d(vn) = 1, d(v′n) = 1 and d(vj) = 2, 1 ≤ j ≤ n− 1.

It is to be noted that, |V (G)| = n+ 3 and |E(G)| = 2n.

Consider a bijection f from V (G) to {1, 2, . . . n+ 3} defined as below.

f(v0) = 1.

f(vn) = 3.

f(v′0) = 2.

f(v′n) = 5.

f(v1) = 4.

f(vj) = 4 + j; 2 ≤ j ≤ n− 1.

By looking into the above prescribed pattern, ef (1) = n = ef (0).

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, the graph constructed by duplicating any edge in K1,n is SDC graph.

Example 6.5.1. The star graph K1,8 and SDC labeling in the graph constructed by

duplicating edge v0v8 in K1,8 are demonstrated in the following Figure 6.44.

12

4

6

7

8 3

11

5v0

v1

v2

v3

v4 v5

v6

v8

v7

v0 v8

v7

v1

v2

v3

v4

v5

v6

v'0v'8

10

9

Figure 6.44: SDC labeling in the graph constructed by duplicating edge in K1,8

184


Theorem 6.5.3. The graph constructed by duplicating a vertex by an edge in star

K1,n is SDC graph.


are pendant vertices of K1,n.

E(K1,n) = {v0vj | 1 ≤ j ≤ n}.Let G denote the graph constructed by duplicating a vertex vj by edge v′jv

′′j in K1,n.



Case 1: deg(v0) = n.

Let us duplicate apex vertex v0 by an edge v′0v′′0 .

It is to be noted that, |V (G)| = n+ 3 and |E(G)| = n+ 3.

Consider a bijection f from V (G) to {1, 2, 3, . . . , n+ 3} defined as below.

f(v0) = 1.

f(v1) = 3.

f(v′0) = 4.

f(v′′0) = 2.

f(vj) = 3 + j; 2 ≤ j ≤ n.



n ≡ 0, 2(mod 4) ef (1) =⌊n+32

⌋, ef (1) =

⌈n+32

⌉

n ≡ 1, 3(mod 4) ef (1) = n+32

= ef (0)

Case 2: deg(vj) = 1.

Let us duplicate pendant vertex vj by an edge v′jv′′j .

WLOG, assume that vj = vn. Then the vertices vn, v′n and v′′n produce a cycle in G.


Consider a bijection f from V (G) to {1, 2, 3, . . . n+ 2, n+ 3} defined as below.

f(v0) = 1.

f(vn) = 3.

185


f(v′n) = 4.

f(v′′n) = 2.

f(vj) = 4 + j; 1 ≤ j ≤ n− 1.



n ≡ 0, 2(mod 4) ef (1) =⌊n+32

⌋, ef (1) =

⌈n+32

⌉

n ≡ 1, 3(mod 4) ef (1) = n+32

= ef (0)


That is, the graph constructed by duplicating vertex by edge in K1,n is SDC.


duplicating apex vertex v0 by edge v′0v′′0 in K1,5 are demonstrated in the following

Figure 6.45.

v0

v0'

v0"

v1

v2

v3

v4

v5

1

3

2

5

4

7

6

8

v0

v1

v2

v3 v4

v5

Figure 6.45: SDC labeling in the graph constructed by duplicating apex vertex v0 by edge v′0v′′0 in

K1,5


duplicating vertex v7 by edge v′7v′′7 in K1,7 are demonstrated in the following Figure

6.46.

v4 v5

v6

v7

v0

v1

v2

v3

v4 v5

v6

v7 v7'

v7"

1

3

2

5

4

7

6

8 9

10

v0

v1

v2

v3

Figure 6.46: SDC labeling in the graph constructed by duplicating vertex v7 by edge v′7v′′7 in K1,7

186


Theorem 6.5.4. The graph constructed by duplicating any arbitrary edge by a vertex

in star K1,n is SDC graph.


are pendant vertices.

E(K1,n) = {v0vj | 1 ≤ j ≤ n}.Let G denote the graph constructed by duplicating an edge say v0vn by a vertex v′n.


Consider a bijection f from V (G) to {1, 2, . . . , n+ 2} defined as below.

f(v0) = 1.

f(v1) = 2.

f(v′1) = 4.

f(vj) = 2j − 1. 2 ≤ j ≤ 3

f(vj) = 2 + j; 4 ≤ j ≤ n.



n ≡ 0, 2(mod 4) ef (1) = n+22

= ef (0)

n ≡ 1, 3(mod 4) ef (1) =⌊n+22

⌋, ef (1) =

⌈n+22

⌉

Thus |ef (0)− ef (1)| ≤ 1.

That is, the graph constructed by duplicating edge by vertex in K1,n is SDC.

Example 6.5.4. The star graph K1,6 and SDC labeling in the graph constructed

by duplicating the edge v0v1 by vertex v′1 in K1,6 are demonstrated in the following

Figure 6.47.

187

6.6. SDC Labeling by Duplicating Vertex/Edge in Cycle Graph

v0

v'1

v1

v2

v3

v4

v5

1

3

2

5

4

7

6

8

v6

v0

v1v2

v3

v4 v5

v6

Figure 6.47: SDC labeling in the graph constructed by duplicating an edge by a vertex in K1,6

6.6 SDC Labeling by Duplicating Vertex/Edge in Cycle Graph

Vaidya and Barasara[54] derived some attractive results on geometric mean label-

ing of graphs constructed by duplicating the graph elements. In this segment we

demonstrate some divisor cordial graphs constructed by duplicating vertex/edge in

cycle Cn.

Theorem 6.6.1. The graph constructed by duplicating an arbitrary vertex in cycle

Cn is SDC graph.

Proof. Let V (Cn) = {vj | 1 ≤ j ≤ n}.E(Cn) = {vjvj+1, vnv1 | 1 ≤ j ≤ n− 1}.WLOG, let us assume that vertex v1 is duplicated by vertex v′1.

Also it is to be noted that, |V (G)| = n+ 1 and |E(G)| = n+ 2.

Consider a bijection f from V (G) to {1, 2, 3, . . . n+ 1} defined as below.

f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.

f(v′1) =

n+ 1 ; if n ≡ 0, 1, 3(mod 4).

n ; if n ≡ 2(mod 4).


188



n ≡ 0, 2(mod 4) ef (1) = n+22

= ef (0)

n ≡ 1, 3(mod 4) ef (1) =⌊n+22

⌋, ef (0) =

⌈n+22

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, the graph constructed by duplicating an arbitrary vertex in Cn is SDC.

Example 6.6.1. The cycle graph C5 and SDC labeling in the graph constructed by

duplicating vertex v1 by vertex v′1 in C5 are demonstrated in the following Figure

6.48.

v5

v4v3

v2

v11

3

2 4

5

6v1'

v2

v3v4

v5

v1

Figure 6.48: SDC labeling in the graph constructed by duplicating a vertex in C5

Theorem 6.6.2. The graph constructed by duplicating an arbitrary edge in cycle

Cn is SDC graph.

Proof. Let V (Cn) = {vj | 1 ≤ j ≤ n}.E(Cn) = {vjvj+1, vnv1 | 1 ≤ j ≤ n− 1}.WLOG, let us assume that v1v2 be the duplicated edge.

By the effect of this duplication, let v′1 and v′2 be lately inserted vertices such that

N(v′1) = {v′2, vn} and N(v′2) = {v′1, v3}.Also it is to be noted that, |V (G)| = n+ 2 and |E(G)| = n+ 3.


Case 1: n ≡ 0(mod 4).

f(v′1) = n+ 1.

f(v′2) = n+ 2.

189


f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.

Case 2: n ≡ 2(mod 4).

f(v′1) = n+ 2.

f(v′2) = n.

f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n− 1.

f(vn) = n+ 1.

Case 3: n ≡ 1, 3(mod 4).

f(v′1) = 1.

f(v′2) = 3.

f(vj) =

j + 1 ; j ≡ 1(mod 4)

j + 2 ; j ≡ 2, 3(mod 4)

j + 3 ; j ≡ 0(mod 4); 1 ≤ j ≤ n− 2.

f(vn−1) =

n+ 1 ; if n ≡ 1(mod 4).

n+ 2 ; if n ≡ 3(mod 4).

f(vn) =

n+ 2 ; if n ≡ 1(mod 4).

n+ 1 ; if n ≡ 3(mod 4).



n ≡ 0, 2(mod 4) ef (0) =⌊n+32

⌋, ef (1) =

⌈n+32

⌉

n ≡ 1, 3(mod 4) ef (1) = n+32

= ef (0)

190


Then we get, |ef (0)− ef (1)| ≤ 1.

That is, the graph constructed by duplicating an arbitrary edge in Cn is SDC.


duplicating edge v1v2 in C6 are demonstrated in the following Figure 6.49.

v6

v5

v4

v1

v2

v3

3

12

4

5

7

6

8

v2'

v1'

v6

v5

v4

v3

v2

v1

Figure 6.49: SDC labeling in the graph constructed by duplicating an edge in C6

Theorem 6.6.3. The graph constructed by duplicating any arbitrary vertex by a

new edge in cycle Cn is SDC graph for n ≡ 0, 1, 2(mod 4).

Proof. Let V (Cn) = {vj | 1 ≤ j ≤ n} and E(Cn) = {vjvj+1, vnv1 | 1 ≤ j ≤ n− 1}.WLOG, let v1 be the vertex duplicated by edge v′1v

′2.



f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.

f(v′1) =

n+ 1 ; if n ≡ 0, 1(mod 4).

n ; if n ≡ 2(mod 4).

f(v′2) = n+ 2.


191



n ≡ 0(mod 4) ef (1) =⌊n+32

⌋, ef (0) =

⌈n+32

⌉

n ≡ 1(mod 4) ef (1) = n+32

= ef (0)

n ≡ 2(mod 4) ef (0) =⌊n+32

⌋, ef (1) =

⌈n+32

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, the graph constructed by duplicating any arbitrary vertex by a new edge

in K1,n is SDC.


duplicating vertex v1 by new edge v′1v′2 in C5 are demonstrated in the following Figure

6.50.

v1

v5

v4v3

v2

v1' v2'

1

3

2 4

5

6 7

v1

v5

v4v3

v2

Figure 6.50: SDC labeling in the graph constructed by duplicating vertex by edge in C5

Theorem 6.6.4. The graph constructed by duplicating each vertex by new edge in

cycle Cn is SDC graph.

Proof. Let V (Cn) = {vj | 1 ≤ j ≤ n}.E(Cn) = {vjvj+1, vnv1 | 1 ≤ j ≤ n− 1}.Let G denote the graph constructed by duplicating each vertex by new edge in cycle

Cn.

Also it is to be noted that, |V (G)| = 3n and |E(G)| = 4n.

Consider a bijection f from V (G) to {1, 2, 3, . . . 3n} defined as below.

192


For n ≡ 0, 1, 3(mod 4) :

f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.

Case 1: n ≡ 0, 3(mod 4).

f(v′j) =

2j − 1 + n ; j ≡ 1, 3(mod 4)

2j − 2 + n ; j ≡ 0, 2(mod 4); 1 ≤ j ≤⌈n2

⌉.

f(v′′j ) = f(v′j) + 2; 1 ≤ j ≤⌈n

2

⌉.

f(v′j) = n+ 2j − 1;⌈n

2

⌉+ 1 ≤ j ≤ n.

f(v′′j ) = n+ 2j;⌈n

2

⌉+ 1 ≤ j ≤ n.

Case 2: n ≡ 1(mod 4).

f(v′j) =

2j + n ; j ≡ 1, 3(mod 4)

2j − 3 + n ; j ≡ 0, 2(mod 4); 1 ≤ j ≤⌊n2

⌋.

f(v′′j ) = f(v′j) + 2; 1 ≤ j ≤⌊n

2

⌋.

f(v′j) = n+ 2j − 1;⌊n

2

⌋+ 1 ≤ j ≤ n.

f(v′′j ) = n+ 2j;⌊n

2

⌋+ 1 ≤ j ≤ n.

Case 3: n ≡ 2(mod 4).

f(vj) =

j + 1 ; j ≡ 1(mod 4)

j + 2 ; j ≡ 2(mod 4)

j − 2 ; j ≡ 3(mod 4)

j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n.

f(v′1) = n− 1.

f(v′′1) = n+ 1.

193


f(v′j) =

2j − 3 + n ; j ≡ 1, 3(mod 4)

2j + n ; j ≡ 0, 2(mod 4); 1 ≤ j ≤ n2.

f(v′′j ) = f(v′j) + 2; 2 ≤ j ≤ n

2.

f(v′j) = n+ 2j − 1;n

2+ 1 ≤ j ≤ n.

f(v′′j ) = n+ 2j;n

2+ 1 ≤ j ≤ n.

Due to the above prescribed pattern, we have

ef (1) = 2n = ef (0).

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, the graph constructed by duplicating each vertex by new edge in cycle Cn

is SDC.


duplicating each vertex by new edge in cycle C5 are demonstrated in the following

Figure 6.51.

v3

v5

v1

v2

v4

v1'

v5'

v1"

v5"

v4'v3"

v3'

v2"

v2'

v4"

1

3

2 4

56

7

8

11

9

12

13

14

15

10

v3

v5

v1

v2

v4

Figure 6.51: SDC labeling in the graph constructed by duplicating each vertex by new edge incycle C5

Theorem 6.6.5. The graph constructed by duplicating any arbitrary edge by a new

vertex in cycle Cn is SDC graph.

Proof. Let V (Cn) = {vj | 1 ≤ j ≤ n} and E(Cn) = {vjvj+1, vnv1 | 1 ≤ j ≤ n− 1}.WLOG, let v1vn be the edge which is duplicated by new vertex v′.

194




Case 1: n ≡ 0, 1, 3(mod 4).

f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.

f(v′) = n+ 1.

Case 2: n ≡ 2(mod 4).

f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n− 1.

f(vn) = n+ 1.

f(v′) = n.



n ≡ 1(mod 4) ef (1) = n+32

= ef (0)

n ≡ 2(mod 4) ef (0) =⌊n+32

⌋, ef (1) =

⌈n+32

⌉

n ≡ 3(mod 4) ef (1) =⌊n+32

⌋, ef (0) =

⌈n+32

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, the graph constructed by duplicating any arbitrary edge by new vertex in

Cn is SDC.


duplicating edge v1v2 by new vertex v′ in C7 are demonstrated in the following Figure

6.52.

195

6.7. SDC Labeling by duplicating Vertex/Edge in Path Graph

v7

v6v5

v4

v3

v2

v1

v'

1

3

2

4 5

6

7

8

v6

v5v4

v3

v2

v1

v7

Figure 6.52: SDC labeling in the graph constructed by duplicating edge by new vertex in C7

6.7 SDC Labeling by duplicating Vertex/Edge in Path Graph

In the previous segment, SDC graphs using duplicating vertex/edge in cycle Cn

are derived while in this segment we demonstrate some divisor cordial graphs con-

structed by duplicating vertex/edge in path Pn.

Theorem 6.7.1. The graph constructed by duplicating an arbitrary vertex in path

Pn is SDC graph.

Proof. Let V (Pn) = {vj | 1 ≤ j ≤ n}, where v1, vn are pendant vertices and

vj(2 ≤ j ≤ n− 1) are internal vertices.

To label the vertices in the graph constructed by duplicating vertex in Pn, we need

to consider the following two cases.

Corresponding to the vertices of different degree in Pn, it is required to discuss



WLOG let the pendant vertex v1 be duplicated by new vertex v′1.

Also it is to be noted that, |V (G)| = n+ 1 and |E(G)| = n.

Consider a bijection f from V (G) to {1, 2, 3, . . . , n+ 1} as follows.

f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.

f(v′1) = n+ 1.

196


Case 2: deg(vk) = 2.

WLOG let v3 be the vertex of degree 2 which is duplicated by new vertex v′3.



f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.

f(v′3) = n+ 1.



n ≡ 0(mod 4) ef (0) =⌊n2

⌋, ef (1) =

⌈n2

⌉

n ≡ 1, 2, 3(mod 4) ef (1) =⌊n2

⌋, ef (0) =

⌈n2

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, the graph constructed by duplicating any arbitrary vertex in Pn is SDC.

Example 6.7.1. SDC labeling in the graph constructed by duplicating pendant vertex

v1 and vertex v3 (of degree 2) in P5 are demonstrated in the following Figure 6.53.

v1 v2 v3 v4 v5

1 3 2 4 5

6 v3'

v1 v2 v3 v4 v5

1 3 2 4 5

6 v1'

Figure 6.53: SDC labeling in the graph constructed by duplicating a vertex in P5

Theorem 6.7.2. The graph constructed by duplicating an arbitrary edge in path Pn

is SDC graph.



197


Let E(Pn) = {vjvj+1 | 1 ≤ j ≤ n− 1}.In Pn, there are two types of edges.

(1) Pendant edges (whose one end vertex is pendant vertex).

(2) Internal edges.

To label the vertices in the graph constructed by duplicating edge in Pn, we need to

consider following two cases.

Case 1: Duplicating a pendant edge.

WLOG, let v1v2 be the duplicated edge.

Let v′1 and v′2 be lately inserted vertices due to the duplicating such that N(v′1) =

{v′2} and N(v′2) = {v′1, v3}.Also it is to be noted that, |V (G)| = n+ 2 and |E(G)| = n+ 1.


f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4); 1 ≤ i ≤ n.

f(v′1) = n+ 1.

f(v′2) = n.

Case 2: Duplicating an internal edge.

Let v2v3 be duplicated internal edge.

Let v′2 and v′3 be lately inserted vertices due to the duplicating such that N(v′2) =

{v′3, v1} and N(v′3) = {v′2, v4}.Also it is to be noted that, |V (G)| = n+ 2 and |E(G)| = n+ 2.


f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4); 1 ≤ i ≤ n.

f(v′2) = n+ 1.

f(v′3) = n.

198




n ≡ 0(mod 4) ef (0) =⌊n+12

⌋, ef (1) =

⌈n+12

⌉

n ≡ 1, 2, 3(mod 4) ef (1) =⌊n+12

⌋, ef (0) =

⌈n+12

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, the graph constructed by duplicating an arbitrary edge in Pn is SDC.

Example 6.7.2. SDC labeling in the graph constructed by duplicating pendant edge

v1v2 and duplicating internal edge v2v3 in P5 are demonstrated in the following Figure

6.54.

v1 v2 v3 v4 v5

1 3 2 4 5

7 6

v1 v2 v3 v4 v5

1 3 2 4 5

7 6

Figure 6.54: SDC labeling in the graph constructed by duplicating edge in P5

Theorem 6.7.3. The graph constructed by duplicating an arbitrary vertex by a new

edge in path Pn is SDC graph.



E(Pn) = {vjvj+1 | 1 ≤ j ≤ n− 1}.WLOG, let vk be the vertex duplicated by edge v′kv

′′k .

Let v′k and v′′k be lately inserted vertices due to the duplicating such that N(v′k) =

{v′′k , vk} and N(v′′k) = {v′k, vk}.Also it is to be noted that, |V (G)| = n+ 2 and |E(G)| = n+ 2.

199



f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.

f(v′k) = n.

f(v′′k) = n+ 1.



n ≡ 0, 2(mod 4) ef (0) = n+22

= ef (1)

n ≡ 1, 3(mod 4) ef (1) =⌊n+22

⌋, ef (0) =

⌈n+22

⌉

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, the graph constructed by duplicating an arbitrary vertex in Pn is SDC.

Example 6.7.3. SDC labeling in the graph constructed by duplicating vertex v′2 by

a new edge v′2v′′2 in P5 is demonstrated in the following Figure 6.55.

v1 v2 v3 v4 v5

1 3 2 4 5

v2' v2"

76

Figure 6.55: SDC labeling in the graph constructed by duplicating vertex v′2 by a new edge v′2v′′2

in P5

Theorem 6.7.4. The graph constructed by duplicating an arbitrary edge by a new

vertex in path Pn is SDC graph.



E(Pn) = {vjvj+1 | 1 ≤ j ≤ n− 1}.WLOG, let vkvk+1 be the duplicated edge and v′ be the lately inserted vertex due

to this duplicating.


200



f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.

f(v′) = n+ 1.



n ≡ 0, 2(mod 4) ef (0) =⌊n+12

⌋, ef (1) =

⌈n+12

⌉

n ≡ 1, 3(mod 4) ef (0) = n+12

= ef (1)

Then we get, |ef (0)− ef (1)| ≤ 1.

That is, the graph constructed by duplicating an arbitrary edge by a new vertex in

Pn is SDC.

Example 6.7.4. SDC labeling in the graph constructed by duplicating edge v2v3 by

a new vertex v′ in P5 is demonstrated in the following Figure 6.56.

v1 v2 v3 v4 v5

1 3 2 4 5

6

Figure 6.56: SDC labeling in the graph constructed by duplicating an edge v2v3 by a new vertexv′ in P5

Theorem 6.7.5. The graph constructed by duplicating each vertex by edge in path

Pn is SDC graph.



E(Pn) = {vjvj+1 | 1 ≤ j ≤ n− 1}.Let G denote the graph constructed by duplication each vertex by edge in path Pn.

Also it is to be noted that, |V (G)| = 3n and |E(G)| = 4n− 1.

Let the edge inserted due to duplicating vertex vk has end vertices v′k and v′′k .

201


Consider a bijection f from V (G) to {1, 2, 3, . . . 3n} defined as below.

For n ≡ 0, 1, 3(mod 4) :

f(vj) =

j ; j ≡ 0, 1(mod 4)

j + 1 ; j ≡ 2(mod 4)

j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.

For the remaining vertices v′1, v′2, . . . , v

′n, v

′′1 , v′′2 , . . . , v

′′n, let us consider following cases.

Case 1: n ≡ 0, 3(mod 4).

f(v′j) =

2j − 1 + n ; j ≡ 1, 3(mod 4)

2j − 2 + n ; j ≡ 0, 2(mod 4); 1 ≤ j ≤⌈n2

⌉.

f(v′′j ) = f(v′j) + 2, 1 ≤ j ≤⌈n

2

⌉.

f(v′j) = n+ 2j − 1;⌈n

2

⌉+ 1 ≤ j ≤ n.

f(v′′j ) = n+ 2j;⌈n

2

⌉+ 1 ≤ j ≤ n.

Case 2: n ≡ 1(mod 4).

f(v′j) =

2j + n ; j ≡ 1, 3(mod 4)

2j − 3 + n ; j ≡ 0, 2(mod 4); 1 ≤ j ≤⌊n2

⌋.

f(v′′j ) = f(v′j) + 2; 1 ≤ j ≤⌊n

2

⌋.

f(v′j) = n+ 2j − 1;⌊n

2

⌋+ 1 ≤ j ≤ n.

f(v′′j ) = n+ 2j;⌊n

2

⌋+ 1 ≤ j ≤ n.

Case 3: n ≡ 2(mod 4).

f(vj) =

j + 1 ; j ≡ 1(mod 4)

j + 2 ; j ≡ 2(mod 4)

j − 2 ; j ≡ 3(mod 4)

j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n− 1.

202


f(v′j) =

2j − 3 + n ; j ≡ 1, 3(mod 4)

2j + n ; j ≡ 0, 2(mod 4); 1 ≤ j ≤ n2.

f(v′′j ) = f(v′j) + 2; 1 ≤ j ≤ n

2.

f(v′j) = n+ 2j − 1;n

2+ 1 ≤ j ≤ n.

f(v′′j ) = n+ 2j;n

2+ 1 ≤ j ≤ n.



n ≡ 0(mod 4) ef (0) =⌊4n−1

2

⌋, ef (1) =

⌈4n−1

2

⌉

n ≡ 1, 2, 3(mod 4) ef (0) =⌈4n−1

2

⌉, ef (1) =

⌊4n−1

2

⌋

Then we get, in each case |ef (0)− ef (1)| ≤ 1.

That is, the graph constructed by duplicating of each vertex by edge in Pn is SDC.

Example 6.7.5. SDC labeling in the graph constructed by duplicating each vertex

by edge in P5 is demonstrated in the following Figure 6.57.

v1 v2 v3 v4 v5

1 3 2 4 5

7

6

v1' v1"

v2"v2' v4"v4'

v3"v3' v5"v5'

8

119

12 13

14 1510

Figure 6.57: SDC labeling in the graph constructed by duplicating each vertex by edge in P5


In this chapter we have emanated new SDC graphs which are constructed from the

standard graph families by considering graph operations such as ringsum of different

graphs with star graph, switching of vertex in cycle allied graph and duplicating

vertex in star, path and cycle graphs. At the end, we pose some open problems.

203


Problem 6.8.1. Classify/Generalize the graphs G such that G⊕K1,n is SDC graph.

(Here |V (G)| may or may not be equal to n.)

Problem 6.8.2. Investigate some new SDC graphs with respect to other graph op-

erations.

Problem 6.8.3. The graph constructed from switching of any vertex (except apex)

of helm Hn is SDC graph.

Problem 6.8.4. The graph constructed from switching of any vertex (except apex)

of closed helm CHn is SDC graph.

204

CHAPTER 7

Summary

This thesis contributes extensive results, novel graphs and connecting concepts of

graph labeling techniques with other field of mathematics such as combinatorics and

number theory.

Labeling of a graph is a bridge connecting combinatorics and graph theory. Large

number of Number Theory results have been used to prove different graph families

which satisfy different graph labeling patterns. During the entire research work du-

ration, 16 new graphs in the field of divisor cordial labeling, 8 new graphs in the

field of square divisor cordial and cube divisor cordial labeling, 24 new graphs in the

field of vertwx odd divisor cordial labeling and 68 new graphs in the field of sum

divisor cordial labeling are discovered. We have derived some algebraic properties of

it. We have also stated open problems and future scope of research in each labeling

technique at the end of each chapter.

205

Details of papers presented in conferences and pub-lished in journals arising from the thesis

Below is the list of research papers published in journals and presented in con-

ferences to obtain the depth of content and to acquire knowledge about the current

ongoing research.

Research Papers Presented in Conferences:

z Research paper entitled “Divisor Cordial Labeling for Vertex Switching and

Duplication of Special Graphs” was presented in National Conference on Al-

gebra, Analysis & Graph Theory [NCAAG - 2017] during 9-11 February 2017,

organized by Department of Mathematics, Saurastra University, Rajkot.

z Research paper entitled “Sum Divisor Cordial Labeling for Vertex Switching

of Cycle Related graphs” was presented in National Conference on Recent Ad-

vancements in Graph Theory, [RAGT - 2019] during 9-10 November 2019, or-

ganized by Department of Mathematics, Gujarat University, Ahmedabad.

z Research paper entitled “Sum Divisor Cordial Labeling in the Context of Corona

Product of graphs” was presented in National Conference on Mathematical Sci-

ences [NCMS - 2020] during 10-11 January 2020, organized by Department of

Mathematics, College of Arts and Science, Adipur, Kachchh.

Research Papers Published in Journals:

1. G. V. Ghodasara and D. G. Adalja, Divisor Cordial Labeling for Vertex Switch-

ing and Duplication of Special Graphs, International Journal of Mathematics

and its Applications, Volume 4(3A) (2016), 73-80.

2. G. V. Ghodasara and D. G. Adalja, Square Divisor Cordial, Cube Divisor Cor-

dial and Vertex Odd Divisor Cordial Labeling of Graphs, International Journal

of Mathematics Trends and Technology, Volume 39(2) (2016), 118-122.

3. G. V. Ghodasara and D. G. Adalja, Divisor Cordial Labeling in Context of

Ringsum of Graphs, International Journal of Mathematics and Soft Computing,

Volume 7(3) (2017), 23-31.

206

4. G. V. Ghodasara and D. G. Adalja, Vertex Odd Divisor Cordial Labeling for

Vertex Switching of Special Graphs, Global Journal of Pure and Applied Math-

ematics, Volume 9(13) (2017), 5525-5538.

5. D. G. Adalja and G. V. Ghodasara, Vertex Odd Divisor Cordial Labeling of

Ringsum of Different Graphs with Star Graph, Research & Reviews: Discrete

Mathematical Structures, Volume 5(2) (2018), 1-9.

6. D. G. Adalja and G. V. Ghodasara, Some New Sum Divisor Cordial Graphs,

International Journal of Applied Graph Theory, Volume 2(1) (2018), 19-33.

7. D. G. Adalja and G. V. Ghodasara, Sum Divisor Cordial Labeling of Snakes

Related Graphs, Journal of Computer and Mathematical Sciences, Volume 9(7)

(2018), 754-768.

8. D. G. Adalja and G. V. Ghodasara, Sum Divisor Cordial Labeling of Ring Sum

of a Graph With Star Graph, International Journal of Computer Sciences and

Engineering, Volume 6(5) (2018), 1-7.

9. D. G. Adalja and G. V. Ghodasara, Sum Divisor Cordial Labeling in the Context

of Corona Product of Graphs, Journal of Applied Science and Computations,

Volume 5(10) (2018), 1141-1158.

Research Papers Communicated for Publication in Journals:

1. D. G. Adalja and G. V. Ghodasara, Sum Divisor Cordial Labeling in the Context

of Vertex Switching of Graphs, Malaya Journal of Mathematics.

2. D. G. Adalja and G. V. Ghodasara, Sum Divisor Cordial Labeling for Duplica-

tion of Special Graphs, Mathematics Todays.

207

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[41] R. Frucht and F. Harary, On the corona of two graphs, Aequationes Mathemat-icae (Springer), Volume 4(3), (1970), 322-325.

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[43] R. Ponraj, S. S. Narayanan and R. Kala, Difference cordial labeling of graphs,Global Journal of Mathematical Sciences, Theory and Practical, Volume 5(3),(2013), 185-196.

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[52] S. K. Vaidya and N. H. Shah, Cordial Labeling of Snakes, International Journalof Mathematics And Its Applications, Volume 2(3), (2014), 17-27.

[53] S. K. Vaidya and C. M. Barasara, Edge product cordial labeling of graphs, Jour-nal of Mathematics and Computer Science, Volume 2(5), (2012), 1436-1450.

[54] S. K. Vaidya and C. M. Barasara, geometric mean labeling in the context ofduplication of graph elements, International Journal of Math. Sci. and Engg.Appls, Volume 6(6), (2012), 311-319.

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211

Annexure

212

International Journal of Mathematics and Soft ComputingVol.7, No.1 (2017), 23 - 31. ISSN Print : 2249 - 3328

ISSN Online : 2319 - 5215

Divisor cordial labeling in context of ring sum of graphs

G. V. Ghodasara1, D. G. Adalja2

1H. & H. B. Kotak Institute of ScienceRajkot, Gujarat, India.

gaurang [email protected]

2Marwadi Education FoundationRajkot, Gujarat, India.

[email protected]

Abstract

A graph G = (V,E) is said to have a divisor cordial labeling if there is a bijectionf : V (G) → {1, 2, . . . |V (G)|} such that if each edge e = uv is assigned the label 1 iff(u)|f(v) or f(v)|f(u) and 0 otherwise, then the number of edges labeled with 0 and thenumber of edges labeled with 1 differ by at most 1. If a graph has a divisor cordial labeling,then it is called divisor cordial graph. In this paper we derive divisor cordial labeling of ringsum of different graphs.

Keywords: Divisor cordial labeling, ring sum of two graphs.AMS Subject Classification(2010): 05C78.

1 Introduction

By a graph, we mean a simple, finite, undirected graph. For terms not defined here, we referto Gross and Yellen [3]. For standard terminology and notations related to number theory werefer to Burton [4]. Varatharajan et al.[7] introduced the concept of divisor cordial labeling of agraph. The divisor cordial labeling of various types of graphs are presented in [6, 8]. The briefsummary of definitions which are necessary for the present investigation are provided below.

Definition 1.1. A mapping f : V (G) → {0, 1} is called binary vertex labeling of G and f(v)is called the label of the vertex v of G under f .

Notation 1.2. For an edge e = uv , the induced edge labeling f∗ : E(G)→ {0, 1} is given byf∗(e) = |f(u)− f(v)|. Thenvf (i) := number of vertices of G having label i under f .ef (i) := number of edges of G having label i under f∗.

Definition 1.3. A binary vertex labeling f of a graph G is called cordial labeling if |vf (0) −vf (1)| ≤ 1 and |ef (0)− ef (1)| ≤ 1. A graph G is cordial if it admits cordial labeling.

23

24 G. V. Ghodasara and D. G. Adalja

The concept of cordial labeling was introduced by Cahit[1]. The concept was generalized andextended to k−equitable labeling[2]. There are other labeling schemes with minor variations incordial theme such as the product cordial labeling, total product cordial labeling, prime cordiallabeling and divisor cordial labeling. The present work is focused on divisor cordial labeling.

Definition 1.4. Let G = (V,E) be a simple, finite, connected and undirected graph. Abijection f : V → {1, 2, . . . |V |} is said to be divisor cordial labeling if the induced functionf∗ : E → {0, 1} defined by

f∗(e = uv) =

1; if f(u)|f(v) or f(v)|f(u),0; otherwise,

satisfies the condition |ef (0) − ef (1)| ≤ 1. A graph that admits a divisor cordial labeling iscalled a divisor cordial graph.

Definition 1.5. A chord of a cycle Cn is an edge joining two non-adjacent vertices.

Definition 1.6. Two chords of a cycle are said to be twin chords if they form a triangle withan edge of the cycle Cn.For positive integers n and p with 5 ≤ p+ 2 ≤ n,Cn,p is the graph consisting of a cycle Cn withtwin chords with which the edges of Cn form cycle Cp,C3 and Cn+1−p without chords.

Definition 1.7. A cycle with triangle is a cycle with three chords which by themselves form atriangle.For positive integers p, q, r and n ≥ 6 with p + q + r + 3 = n, Cn(p, q, r) denotes a cycle withtriangle whose edges form the edges of cycles Cp+2, Cq+2, Cr+2 without chords.

Definition 1.8. Ring sum of two graphs G1 = (V1, E1) and G2 = (V2, E2) denoted by G1⊕G2,is the graph G1 ⊕G2 = (V1 ∪ V2, (E1 ∪ E2)− (E1 ∩ E2)).

Remark 1.9. Throughout this paper we consider the ring sum of a graph G with K1,n byconsidering any one vertex of G and the apex vertex of K1,n as a common vertex.

2 Main ResultsTheorem 2.1. Cn ⊕K1,n is a divisor cordial graph for all n ∈ N.

Proof: Let V (Cn ⊕K1,n) = V1 ∪ V2, where V1 = {u1, u2, . . . , un} be the vertex set of Cn andV2 = {v = u1, v1, v2, . . . , vn} be the vertex set of K1,n. Here v1, v2, . . . , vn are pendant verticesand v is the apex vertex of K1,n. Also |V (Cn ⊕K1,n)| = |E(Cn ⊕K1,n)| = 2n.

We define a labeling f : V (Cn ⊕K1,n)→ {1, 2, 3, . . . , 2n} as follows:

f(u1) = f(v) = 2, f(v1) = 1,

Divisor cordial labeling in context of ring sum of graphs 25

f(ui) = 2i− 1; 2 ≤ i ≤ n,f(vj) = 2j; 2 ≤ j ≤ n.

According to this pattern the vertices are labeled such that for any edge e = uiui+1 in Cn,f(ui) - f(ui+1), 1 ≤ i ≤ n− 1.

Also f(v1) | f(v) and f(v) | f(vj) for each j, 2 ≤ j ≤ n.Hence ef (1) = ef (0) = n.

Thus the graph Cn ⊕K1,n admits a divisor cordial labeling and hence Cn ⊕K1,n is a divisorcordial graph.

Example 2.2. A divisor cordial labeling of C5 ⊕K1,5 is shown in Figure 1.

Figure 1: A divisor cordial labeling of C5 ⊕K1,5.

Theorem 2.3. The graph G⊕K1,n is a divisor cordial graph for all n ≥ 4, n ∈ N, where G isthe cycle Cn with one chord forming a triangle with two edges of Cn.

Proof: Let G be the cycle Cn with one chord. Let V (G ⊕ K1,n) = V1 ∪ V2, where V1 is thevertex set of G and V2 is the vertex set of K1,n. Let u1, u2, . . . , un be the successive vertices ofCn and e = u2un be the chord of Cn.

The vertices u1, u2, un form a triangle with the chord e. Let v1, v2, . . . , vn be the pendantvertices, v be the apex vertex ofK1,n. Take v = u1. Also |V (G⊕K1,n)| = 2n and |E(G⊕K1,n)| =2n+ 1.

We define a labeling f : V (G⊕K1,n)→ {1, 2, 3, . . . , 2n} as follows:f(u1) = f(v) = 2, f(v1) = 1,f(ui) = 2i− 1; 2 ≤ i ≤ n,f(vj) = 2j; 2 ≤ j ≤ n.

According to this pattern the vertices are labeled such that for any edge e = uiui+1 ∈ G,f(ui) - f(ui+1), 1 ≤ i ≤ n− 1.


Also f(v1) | f(v) and f(v) | f(vj) for each j, 2 ≤ j ≤ n.

Hence ef (1) = n, ef (0) = n+ 1.

Thus |ef (0) − ef (1)| ≤ 1 and the graph G ⊕ K1,n admits divisor cordial labeling. Therefore,G⊕K1,n is a divisor cordial graph.

Example 2.4. A divisor cordial labeling of ring sum of C6 with one chord and K1,6 is shownin Figure 2.

Figure 2: A divisor cordial labeling of ring sum of C6 with one chord and K1,6.

Theorem 2.5. The graph G⊕K1,n is a divisor cordial graph for all n ≥ 5, n ∈ N, where G isthe cycle with twin chords forming two triangles and another cycle Cn−2 with the edges of Cn.

Proof: Let G be the cycle Cn with twin chords, where chords form two triangles and one cycleCn−2. Let V (G⊕K1,n) = V1 ∪ V2.

V1 = {u1, u2, . . . , un} is the vertex set of Cn, e1 = unu2 and e2 = unu3 are the chords of Cn.

V2 = {v = u1, v1, v2, . . . , vn} is the vertex set of K1,n, where v1, v2, . . . , vn are pendant verticesand v = u1 is the apex vertex.

Also |V (G⊕K1,n)| = 2n and |E(G⊕K1,n)| = 2n+ 2.

We define a labeling f : V (G⊕K1,n)→ {1, 2, 3, . . . , 2n} as follows:

f(u1) = f(v) = 2, f(v1) = 3, f(un) = 2n− 1, f(un−1) = 1,f(ui) = 2i+ 1; 2 ≤ i ≤ n− 2,f(vj) = 2j; 2 ≤ j ≤ n.


Also f(v1) | f(v) and f(v) | f(vj) for each j, 2 ≤ j ≤ n. and f(un−1) | f(un−2). Henceef (0) = n+ 1 = ef (1).


Thus the graph admits a divisor cordial labeling and henceG⊕K1,n is a divisor cordial graph.

Example 2.6. A divisor cordial labeling of ring sum of cycle C7 with twin chords and K1,7 isshown in Figure 3.

Figure 3: A divisor cordial labeling of ring sum of C7 with twin chords and K1,7.

Theorem 2.7. The graph G⊕K1,n is a divisor cordial graph for all n ≥ 6, n ∈ N, where G isa cycle with triangle Cn(1, 1, n− 5).

Proof: Let G be cycle with triangle Cn(1, 1, n − 5). Let V (G ⊕ K1,n) = V1 ∪ V2, whereV1 = {u1, u2, . . . , un} is the vertex set of G and V2 = {v = u1, v1, v2, . . . , vn} is the vertex setof K1,n. Here v1, v2, . . . , vn are the pendant vertices and v is the apex vertex of K1,n.

Let u1, u3 and un−1 be the vertices of triangle formed by edges e1 = u1u3, e2 = u3un−1 ande3 = u1un−1. Also |V (G⊕K1,n)| = 2n and |E(G⊕K1,n)| = 2n+ 3.


f(u1 = v) = 2, f(v1) = 3, f(u2) = 1,f(ui) = 2i− 1; 3 ≤ i ≤ n,f(vj) = 2j; 2 ≤ j ≤ n.


Also f(v1) - f(v) and f(v) | f(vj) for each j, 2 ≤ j ≤ n. f(v) | f(u2), f(u2) | f(u3) .

Then |ef (0)− ef (1)| ≤ 1. Hence G⊕K1,n is a divisor cordial graph.

Example 2.8. A divisor cordial labeling of ring sum of cycle C8 with triangle and K1,8 isshown in Figure 4.


Figure 4: A divisor cordial labeling of ring sum of C8(1, 1, 3) and K1,8.

Theorem 2.9. The graph Pn ⊕K1,n is a divisor cordial graph for all n ∈ N.

Proof: Let V (Pn ⊕K1,n) = V1 ∪ V2, where V1 = {u1, u2, . . . , un} is the vertex set of Pn andV2 = {v = u1, v1, v2, . . . , vn} is the vertex set of K1,n. Here v1, v2, . . . , vn are the pendantvertices and v is the apex vertex. Also |V (G⊕K1,n)| = 2n, |E(G⊕K1,n)| = 2n− 1.


f(u1) = f(v) = 2, f(v1) = 1,f(ui) = 2i− 1; 2 ≤ i ≤ n,f(vj) = 2j; 2 ≤ j ≤ n.

According to this pattern the vertices are labeled such that for any edge e = uiui+1 ∈ G,f(ui) - f(ui+1), 1 ≤ i ≤ n− 1.Also f(v) | f(vj) for each j, 1 ≤ j ≤ n.

Then we have ef (1) = n, ef (0) = n − 1. Hence the graph admits a divisor cordial labeling.Theredore, Pn ⊕K1,n is a divisor cordial graph.

Example 2.10. A divisor cordial labeling of P5 ⊕K1,5 is shown in Figure 5.

Figure 5: A divisor cordial labeling of G⊕K1,5.

Definition 2.11. The double fan graph DFn is defined as DFn = Pn + 2K1.

Theorem 2.12. The graph DFn ⊕K1,n is a divisor cordial graph for every n ∈ N.


Proof: Let V (DFn ⊕ K1,n) = V1 ∪ V2, where V1 = {u,w, u1, u2, . . . , un} be the vertex set ofDFn and V2 = {v = w, v1, v2, . . . , vn} be the vertex set of K1,n. Here v1, v2, . . . , vn are pendantvertices and v be the apex vertex of K1,n. Also |V (G⊕K1,n)| = 2n+ 2, |E(G⊕K1,n)| = 4n−1.We define a labeling f : V (G⊕K1,n)→ {1, 2, 3, . . . , 2n+ 2} as follows:

f(w) = f(v) = 1, f(u) = p, where p is largest prime number.f(v1) = 2, f(u1) = 3,f(ui) = 2i; 1 ≤ i ≤ n.

Assign the remaining labels to the remaining vertices v1, v2, . . . , vn in any order.

According to this pattern the vertices are labeled such that for any edge e = uiui+1, f(ui) -f(ui+1) , 1 ≤ i ≤ n − 1. Also f(v) | f(vj) for each j, 1 ≤ j ≤ n. and f(v) | f(ui) for each i,1 ≤ i ≤ n. and f(u) - f(ui) for each i, 1 ≤ i ≤ n.

Then we have ef (1) = 2n, ef (0) = 2n − 1. Hence the graph admits divisor cordial labeling.Therefore, DFn ⊕K1,n is a divisor cordial graph.

Example 2.13. A divisor cordial labeling of DF5 ⊕K1,5 is shown in Figure 6.

Figure 6: A divisor cordial labeling of DF5 ⊕K1,5.

Definition 2.14. The flower fln is the graph obtained from a helm Hn by joining each pendantvertex to the apex of the helm. It contains three types of vertices: an apex of degree 2n, nvertices of degree 4 and n vertices of degree 2.

Theorem 2.15. The graph fln ⊕K1,n is a divisor cordial graph for every n ∈ N.

Proof: Let V (fln⊕K1,n) = V1∪V2, V1 = {u, u1, u2, . . . , un, w1, w2, . . . , wn} be the vertex set offln, where u is the apex vertex, u1, u2, . . . , un are the internal vertices and w1, w2, . . . , wn are theexternal vertices. Let V2 = {v = w1, v1, v2, . . . , vn} be the vertex set ofK1,n, where v1, v2, . . . , vnare pendant vertices and v is the apex vertex of K1,n. Also note that |V (fln ⊕ K1,n)| =3n+ 1, |E(G⊕K1,n)| = 5n.

We define a labeling f : V (fln ⊕K1,n)→ {1, 2, 3, . . . , 3n+ 1} as follows:


f(u) = 1,f(ui) = 2i+ 1; 1 ≤ i ≤ n,f(wi) = 2i; 1 ≤ i ≤ n.

Assign the remaining labels to the remaining vertices v1, v2, . . . , vn in any order. According tothis pattern the vertices are labeled such that for any edge e = uiui+1 ∈ G,f(ui) - f(ui+1) ,1 ≤ i ≤ n − 1. Further f(u) | f(ui), f(u) | f(wi) for each i, 1 ≤ i ≤ n and f(v) | f(vi) if i isodd, 1 ≤ i ≤ n.

Then |ef (0)− ef (1)| ≤ 1. Hence the graph admits divisor cordial labeling and fln ⊕K1,n is adivisor cordial graph.

Example 2.16. A divisor cordial labeling of the graph fl4 ⊕K1,4 is shown in Figure 7.

Figure 7: A divisor cordial labeling of fl4 ⊕K1,4.

Remark 2.17. In all the above theorems, for the ring sum operation one can consider anyarbitrary vertex of G and by different permutations of the vertex labels provided in the abovedefined labeling pattern one can easily check that the resultant graph is divisor cordial.

Concluing Remarks: The divisor cordial labeling is an invariant of cordial labeling by minorvariation in the definition using divisor of a number. Here we have derived divisor cordialgraphs in context of the operation ring sum of graphs. It is interesting to see whether divisorcordial graphs are invariant under ring sum or any other graph operation or not.

References[1] I. Cahit, Cordial Graphs: A weaker version of graceful and harmonious Graphs, Ars Com-

binatoria, 23 (1987), 201-207.

[2] I. Cahit, On cordial and 3-equitable labellings of graphs, Util. Math., 37(1990), 189-198.

[3] J. Gross and J. Yellen, Graph Theory and its applications, CRC Press, 1999.

[4] D. M. Burton, Elementary Number Theory, Brown Publishers, Second Edition, 1990.


[5] J. A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics,19, (2015), # DS6.

[6] S. K. Vaidya and N. H. Shah, Further Results on Divisor Cordial Labeling, Annals of Pureand Applied Mathematics, 4(2) (2013), 150-159.

[7] R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan, Divisor Cordial Graphs, In-ternational J. Math. Combin., 4 (2011), 15-25.

[8] R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan, Special Classes of DivisorCordial Graphs, International Mathematical Forum, 7 (35) (2012), 1737- 1749.

Sum Divisor Cordial Labeling in the Context of Corona

Product of Graphs

D. G. Adalja 1, G. V. Ghodasara 2

1 Marwadi Education Foundation,

Rajkot, Gujarat - INDIA

[email protected]

2 H. & H. B. Kotak Institute of Science,

Rajkot, Gujarat - INDIA

[email protected]

Abstract

A graph G = (V,E) is said to have a sum divisor cordial labeling if there exists a bijection

f : V (G) → {1, 2, 3, . . . , |V (G)|} such that each edge e = uv is assigned the label 1 if 2 divides

[f(u) + f(v)] and 0 otherwise, then the number of edges labeled with 0 and the number of edges

labeled with 1 differ by at most 1. If a graph admits a sum divisor cordial labeling, then it is

called sum divisor cordial graph. In this paper we have derived the graphs obtained by taking

corona product of K1 with different graphs like star K1,n, complete bipartite graphs K2,n and

K3,n, wheel, helm, flower, fan, double fan, barycentric subdivision of the star K1,n, cycle with

one chord, twin chords and triangle admit sum divisor cordial labeling.

Key words: Sum divisor cordial labeling, Corona product of two graphs.

AMS Subject classification number: 05C78.

1 Introduction

In this paper, by a graph, we mean a simple, finite, undirected graph. For terms and notations

related to graph theory which are not defined here, we refer to Gross and Yellen[6] and for standard

terminology and notations related to number theory we refer to Burton[2].

Remark 1.1. Throughout this paper |V (G)| and |E(G)| denote the cardinality of vertex set and edge

set of graph G respectively.

Rosa[9] introduced grpah labeling as follows.

If the vertices or edges or both of the graph are assigned valued subject to certain conditions, then it

is known as graph labeling.

Varatharajan et al. introduced the concept of divisor cordial labeling of a graph. The definition of

divisor cordial labeling is given below.

Definition 1.1 ([10]). Let G = (V,E) be a simple, finite, connected and undirected graph. A bijection

f : V (G)→ {1, 2, . . . , |V (G)|} is said to be divisor cordial labeling if the induced function f∗ : E(G)→{0, 1} defined by

f∗(e = uv) =

1 if f(u) | f(v) or f(v) | f(u);

0 otherwise.

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satisfies the condition |ef (0)− ef (1)| ≤ 1.

A graph with a divisor cordial labeling is called a divisor cordial graph.

In [10], Varatharajan et al. proved that the graphs such as path, cycle, wheel, star and some

complete bipartite graphs are divisor cordial graphs. They have also derived some special classes of

divisor graphs such as full binary tree, dragon, corona, G ∗K2,n and G ∗K3,n.

Ghodasara and Adalja[5] derived divisor cordial labeling for graphs obtained by ring sum of some

standard graphs with star graph.

Motivated through the concept of divisor cordial labeling, A. Lourdusamy, F. Patrick and J.

Shiama introduced the concept of sum divisor cordial labeling of graphs which is defined as follows.

Definition 1.2 ([7]). Let G = (V,E) be a simple graph, f : V (G) → {1, 2, 3, . . . , |V (G)|} be a

bijection and the induced function f∗ : E(G)→ {0, 1} be defined as

f∗(e = uv) =

1 if 2 | [f(u) + f(v)];

0 otherwise.

Then f is called sum divisor cordial labeling if |ef (0)− ef (1)| ≤ 1.

A graph which admits sum divisor cordial labeling is called sum divisor cordial graph.

In [8], Lourdusamy et al. proved that shadow graph and splitting graph of K1,n, shadow graph,

subdivision graph, splitting graph and degree splitting graph of Bn,n, subdivision graph of ladder,

corona of ladder and triangular ladder with K1, closed helm are sum divisor cordial graphs.

In [1], Adalja and Ghodasara derived some more sum divisor cordial graphs.

Definition 1.3 ([3]). The corona of a graph G with another graph H, denoted as G�H, is the graph

obtained by taking one copy of G and |V (G)| copies of H and joining the ith vertex of G with an edge

to every vertex in the ith copy of H.

Definition 1.4 ([6]). Let G = (V,E) be a graph. Let e = uv be an edge of G and w be a vertex not

in G. The edge e is said to be subdivided when it is replaced by the edges e′ = uw and e′′ = wv.

Definition 1.5 ([6]). If every edge of graph a G is subdivided, then the resulting graph is called

barycentric subdivision of graph G. In other words barycentric subdivision of the graph is obtained by

inserting a vertex of degree 2 into every edge of original graph. The barycentric subdivision of any

graph G is denoted by S(G).

2 Results

Theorem 2.1. K1,n �K1 is a sum divisor cordial graph.

Proof. Let v1, v2, . . . , vn be pendant vertices and v0 be the apex vertex of K1,n.

Let v′0, v′1, v′2, . . . , v

′n be the newly added vertices to obtain the graph K1,n �K1.

V (K1,n �K1) = V (K1,n) ∪ {v′0, v′1, v′2, . . . , v′n}.E(K1,n �K1) = E(K1,n) ∪ {viv′i; 0 ≤ i ≤ n}.|V (K1,n �K1)| = 2n + 2.

|E(K1,n �K1)| = 2n + 1.

We define labeling f : V (K1,n �K1)→ {1, 2, 3, . . . , 2n + 2} as follows.

f(v0) = 1;

f(v′0) = 2;

f(vi) = 2i + 1 1 ≤ i ≤ n;

f(v′i) = 2i + 2 1 ≤ i ≤ n.

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In view of the above defined labeling pattern, we have

ef (1) =

⌊2n + 1

2

⌋,

ef (0) =

⌈2n + 1

2

⌉.

Thus |ef (1)− ef (0)| ≤ 1.

Thus the graph under consideration admits sum divisor cordial labeling.

That is, K1,n �K1 is a sum divisor cordial graph.

Example 2.1. Sum divisor cordial labeling of the graph K1,6 � K1 is shown in Figure 1 as an

illustration for Theorem 2.1.

v1 v2 v3 v4 v5 v6

v2'v1' v3' v4' v5' v6'

v0'

v01

6

3

4 8 1210 14

5 1197 13

2

Figure 1

Theorem 2.2. K2,n �K1 is sum divisor cordial graph except for n ≡ 0(mod4).

Proof. Let W = U ∪ V be the bipartition of vertex set of K2,n, where

U = {u1, u2} and V = {v1, v2, . . . , vn}.Let u′1, u

′2, v′1, v′2, . . . , v


V (K2,n �K1) = V (K2,n) ∪ {u′1, u′2} ∪ {v′1, v′2, . . . , v′n}.E(K2,n �K1) = E(K2,n) ∪ {u1u

′1, u2u

′2} ∪ {viv′i; 1 ≤ i ≤ n}.

|V (K2,n �K1)| = 2n + 4.

|E(K2,n �K1)| = 3n + 2.


f(u1) = 1;

f(u2) = 2;

f(vi) = i + 2 1 ≤ i ≤ n.

For n ≡ 1(mod4)

f(u′1) = 2n + 3;

f(u′2) = 2n + 4.

For k = n−12 :

f(v′2i−1) = (n + 2) + (2i− 1) 1 ≤ i ≤ k;

f(v′2i) = (n + 1) + 2i 1 ≤ i ≤ k;

f(v′i) = (n + 2) + i k + 1 ≤ i ≤ n.

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For n ≡ 3(mod4)

f(u′1) = 2n + 4;

f(u′2) = 2n + 3.

For k = n+12 :

f(v′2i−1) = (n + 2) + (2i− 1) 1 ≤ i ≤ k;

f(v′2i) = (n + 1) + 2i 1 ≤ i ≤ k;

f(v′i) = (n + 2) + i k + 1 ≤ i ≤ n.

For n ≡ 2(mod4)

f(u′1) = 2n + 3;

f(u′2) = 2n + 4.

For k = n+22 :

f(v′2i−1) = (n + 2) + 2i 1 ≤ i ≤ k;

f(v′2i) = (n + 1) + (2i− 1) 1 ≤ i ≤ k;

f(v′i) = (n + 2) + i k + 1 ≤ i ≤ n.

In view of the above defined labeling pattern we have the following.


n ≡ 1(mod 4) ef (1) =⌈3n+2

2

⌉, ef (0) =

⌊3n+2

2

⌋

n ≡ 2(mod 4) ef (1) = 3n+22 = ef (0)

n ≡ 3(mod 4) ef (1) =⌊3n+2

2

⌋, ef (0) =

⌈3n+2

2

⌉

Thus |ef (1)− ef (0)| ≤ 1.

So, K2,n �K1 is a sum divisor cordial graph.

Example 2.2. Sum divisor cordial labeling of K2,5 �K1 is shown in Figure 2 as an illustration for

Theorem 2.2.

v1 v2 v3 v4 v5

v2'v1' v3' v4' v5'

u1'

u2

u2'

u1

3 5

119

7

12

1 2

8

64

10

13 14

Figure 2

Theorem 2.3. K3,n �K1 is sum divisor cordial graph.

Proof. Let W = U ∪ V be the bipartition of vertex set of K2,n, where

U = {u1, u2, u3} and V = {v1, v2, . . . , vn}.4



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Let u′1, u′2, u′3, v′1, v′2, . . . , v


V (K3,n �K1) = V (K3,n) ∪ {u′j , v′i; 1 ≤ j ≤ 3, 1 ≤ i ≤ n}.E(K3,n �K1) = E(K3,n) ∪ {uju

′j , viv

′i; 1 ≤ j ≤ 3, 1 ≤ i ≤ n}.

|V (K3,n �K1)| = 2n + 6.

|E(K3,n �K1)| = 4n + 3.


f(ui) = i 1 ≤ i ≤ 3;

f(u′1) = 5;

f(u′2) = 4;

f(u′3) = 6;

f(vi) = 6 + (2i− 1) 1 ≤ i ≤ n;

f(v′i) = 6 + (2i) 1 ≤ i ≤ n.

In view of the above defined labeling pattern we have

ef (1) =

⌈4n + 3

2

⌉,

ef (0) =

⌊4n + 3

2

⌋.

Thus |ef (1)− ef (0)| ≤ 1.

So, K3,n �K1 is a sum divisor cordial graph.

Example 2.3. Sum divisor cordial labeling of K3,7 �K1 is shown in Figure 3 as an illustration for

Theorem 2.3.

v1 v2 v3 v4 v5

v2'v1' v3' v4' v5'

u1'

u2

u2'

u1

v6 v7

v6' v7'

u3

u3'64

8 12 14

1 3

5

1197 13

2

1715 19

1816 2010

Figure 3

Definition 2.1 ([6]). The wheel graph is join of K1 and Cn, denoted as Wn = Cn + K1. The cycle

Cn forms rim, the vertices corresponding rim are called rim vertices and the vertex corresponding to

K1 is called apex (or hub).

Theorem 2.4. Wn �K1 is sum divisor cordial graph.

Proof. Let v0 be the apex vertex and v1, v2, . . . , vn be rim vertices of Wn.

Let v′0, v′1, v′2, . . . , v

′n be the newly added vertices to obtain the graph Wn �K1.

V (Wn �K1) = V (Wn) ∪ {v′i; 0 ≤ i ≤ n}.E(Wn �K1) = E(Wn) ∪ {viv′i; 0 ≤ i ≤ n}.|V (Wn �K1)| = 2n + 2.

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|E(Wn �K1)| = 3n + 1.

We define labeling f : V (Wn �K1)→ {1, 2, 3, . . . , 2n + 2} as follows.

For n ≡ 0, 2(mod4)

f(v0) = 1;

f(v′0) = 2n + 2;

f(v2i−1) = 4i− 2 1 ≤ i ≤ n

2;

f(v2i) = 4i− 1 1 ≤ i ≤ n

2;

f(v′2i−1) = 4i 1 ≤ i ≤ n

2;

f(v′2i) = 4i + 1 1 ≤ i ≤ n

2.

For n ≡ 1, 3(mod4)

f(v0) = 2n + 2;

f(v′0) = 1;

f(v2i−1) = 4i− 1 1 ≤ i ≤ n− 1

2;

f(v2i) = 4i− 2 1 ≤ i ≤ n− 1

2;

f(v′2i−1) = 4i + 1 1 ≤ i ≤ n− 1

2;

f(v′2i) = 4i 1 ≤ i ≤ n− 1

2;

f(vn) = 2n;

f(v′n) = 2n + 1.

In view of above defined labeling pattern we have the following.


n ≡ 0, 2(mod 4) ef (1) =⌊3n+1

2

⌋, ef (0) =

⌈3n+1

2

⌉

n ≡ 1, 3(mod 4) ef (1) = 3n+12 = ef (0)

Thus |ef (0)− ef (1)| ≤ 1.

Hence Wn �K1 is a sum divisor cordial graph.

Example 2.4. Sum divisor cordial labeling of W7 �K1 is shown in Figure 4 as an illustration for

Theorem 2.4.

v0

v1

v2

v3

v4

v5

v2'

v1'

v3'

v4'

v5'

v6

v7

v6'

v7'

v0'

1

2

34

5

6

7

8

9

10

11

12

13

14

16

15

Figure 4

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Definition 2.2 ([6]). The helm Hn is the graph obtained from a wheel Wn by attaching a pendant

edge to each rim vertex. It contains three types of vertices: an apex of degree n, n vertices of degree

4 and n pendant vertices.

Theorem 2.5. Hn �K1 is sum divisor cordial graph.

Proof. Let v0 be the apex vertex and v1, v2, . . . , vn be vertices of degree 4 and u1, u2, . . . , un be the

pendant vertices of the helm Hn.

Let v′0, v′1, v′2, . . . , v

′n, u′1, u′2, . . . , u

′n be the newly added vertices to obtain the graph Hn �K1.

V (Hn �K1) = V (Hn) ∪ {v′0, v′i, u′i; 1 ≤ i ≤ n}.E(Hn �K1) = E(Hn) ∪ {v0v′0, viv′i, uiu

′i; 1 ≤ i ≤ n}.

Hence |V (Hn �K1)| = 4n + 2 and |E(Hn �K1)| = 5n + 1.

We define labeling f : V (Hn �K1)→ {1, 2, 3, . . . , 4n + 2} as follows.

For n ≡ 0, 1(mod4) :

f(vi) =

i + 1 i ≡ 0, 1(mod 4);

i + 2 i ≡ 2(mod 4);

i i ≡ 3(mod 4); 1 ≤ i ≤ n.

f(v0) = 1;

f(v′0) = 4n + 2;

f(v′i) = (n + 1) + 2i 1 ≤ i ≤ n;

f(ui) = n + 2i 1 ≤ i ≤ n;

f(u′i) = (3n + 1) + i 1 ≤ i ≤ n.

For n ≡ 2(mod4) :

f(vi) =

i i ≡ 0, 1(mod 4);

i + 1 i ≡ 2(mod 4);

i− 1 i ≡ 3(mod 4); 1 ≤ i ≤ n.

f(v′i) = (n + 1) + 2i 2 ≤ i ≤ n;

f(v′1) = n + 2;

f(u1) = n;

f(ui) = n + 2i 2 ≤ i ≤ n;

f(u′i) = (3n + 1) + i 1 ≤ i ≤ n.

For n ≡ 3(mod4) :

f(vi) =

i + 1 i ≡ 0, 1(mod 4);

i + 2 i ≡ 2(mod 4);

i i ≡ 3(mod 4); 1 ≤ i ≤ n.

f(v0) = 4n + 2;

f(v′0) = 1;

f(v′i) = (n + 1) + 2i 1 ≤ i ≤ n;

f(ui) = n + 2i 1 ≤ i ≤ n;

f(u′i) = (3n + 1) + i 1 ≤ i ≤ n.


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n ≡ 0, 2(mod 4) ef (1) =⌊5n+1

2

⌋, ef (0) =

⌈5n+1

2

⌉

n ≡ 1, 3(mod 4) ef (1) = 5n+12 = ef (0)

Thus |ef (0)− ef (1)| ≤ 1.

Hence Hn �K1 is a sum divisor cordial graph.

Example 2.5. Sum divisor cordial labeling of H7 �K1 is shown in Figure 5 as an illustration for

Theorem 2.5.

u1u7

u7'

v0

v1

v1'

v2

v2'

v3

v3'v4

v4'

v5v5'

v6

v6'v7

v7'

u2

u2'

u1'

u3

u4

u4'

u5 u3'

u5'

u6u6'

v0'1

3

2

4

5

7

6

8

910

11

12

14

13

15

19

16

17

22

18

20

21

23

26

24

25

27

28

29

30

Figure 5

Definition 2.3 ([4]). The flower graph fln(n ≥ 3) is obtained from helm Hn by joining each pendant

vertex to the central vertex of Hn.

It contains three types of vertices: an apex of degree 2n, n vertices of degree 4 and n vertices of degree

2.

Theorem 2.6. fln �K1 is sum divisor cordial graph.

Proof. Let v0 be the apex vertex and v1, v2, . . . , vn be the vertices of degree 4 and u1, u2, . . . , un be

the vertices of degree 2 in the flower fln.

Let v′0, v′1, v′2, . . . , v

′n, u′1, u′2, . . . , u

′n be the newly added vertices to obtain the graph fln �K1.

V (fln �K1) = V (fln) ∪ {v′0, v′i, u′i, ; 1 ≤ i ≤ n},E(fln �K1) = E(fln) ∪ {v0v′0, viv′i, uiu

′i; 1 ≤ i ≤ n}.

Hence |V (fln �K1)| = 4n + 2 and |E(fln �K1)| = 6n + 1.

We define labeling f : V (fln �K1)→ {1, 2, 3, . . . , 4n + 2} as follows.

f(vi) =

i + 1 i ≡ 0, 1(mod 4);

i + 2 i ≡ 2(mod 4);

i i ≡ 3(mod 4); 1 ≤ i ≤ n.

f(v0) = 1;

f(v′0) = 4n + 2;

8



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For n ≡ 0(mod4)

f(v′4i−3) = 3n + 4i− 1 1 ≤ i ≤ n

4;

f(v′4i−2) = 3n + 4i + 1 1 ≤ i ≤ n

4;

f(v′4i−1) = 3n + 4i− 2 1 ≤ i ≤ n

4;

f(v′4i) = 3n + 4i 1 ≤ i ≤ n

4;

f(u2i−1) = n + 4i− 2 1 ≤ i ≤ n

2;

f(u2i) = n + 4i− 1 1 ≤ i ≤ n

2;

f(u′2i−1) = n + 4i 1 ≤ i ≤ n

2;

f(u′2i) = n + 4i + 1 1 ≤ i ≤ n

2.

For n ≡ 2(mod4)

f(v′4i−3) = 3n + 4i− 3 1 ≤ i ≤ n

4;

f(v′4i−2) = 3n + 4i− 1 1 ≤ i ≤ n

4;

f(v′4i−1) = 3n + 4i 1 ≤ i ≤ n

4;

f(v′4i) = 3n + 4i + 2 1 ≤ i ≤ n

4;

f(u2i−1) = n + 4i− 3 1 ≤ i ≤ n

2;

f(u2i) = n + 4i 1 ≤ i ≤ n

2;

f(u′2i−1) = n + 4i− 1 1 ≤ i ≤ n

2;

f(u′2i) = n + 4i + 2 1 ≤ i ≤ n

2.

For n ≡ 1(mod4)

f(v′4i−3) = 3n + 4i 1 ≤ i ≤ n

4;

f(v′4i−2) = 3n + 4i + 2 1 ≤ i ≤ n

4;

f(v′4i−1) = 3n + 4i− 3 1 ≤ i ≤ n

4;

f(v′4i) = 3n + 4i− 1 1 ≤ i ≤ n

4;

f(u2i−1) = n + 4i− 2 1 ≤ i ≤ n + 1

2;

f(u′2i−1) = n + 4i 1 ≤ i ≤ n + 1

2;

f(u2i) = n + 4i− 1 1 ≤ i ≤ n− 1

2;

f(u′2i) = n + 4i + 1 1 ≤ i ≤ n− 1

2;

f(v′n) = 4n.

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For n ≡ 3(mod4)

f(v′4i−3) = 3n + 4i 1 ≤ i ≤ n + 1

4;

f(v′4i−2) = 3n + 4i + 2 1 ≤ i ≤ n

4;

f(v′4i−1) = 3n + 4i− 3 1 ≤ i ≤ n

4;

f(v′4i) = 3n + 4i− 1 1 ≤ i ≤ n

4;

f(u2i−1) = n + 4i− 2 1 ≤ i ≤ n + 1

2;

f(u′2i−1) = n + 4i 1 ≤ i ≤ n + 1

2;

f(u2i) = n + 4i− 1 1 ≤ i ≤ n− 1

2;

f(u′2i) = n + 4i + 1 1 ≤ i ≤ n− 1

2;

f(v′n−1) = 4n− 2;

f(v′n) = 4n.


ef (1) =

⌈6n + 1

2

⌉,

ef (0) =

⌊6n + 1

2

⌋.

Thus |ef (0)− ef (1)| ≤ 1.

Hence fln �K1 admits sum divisor cordial labeling and hence it is a sum divisor cordial graph.

Example 2.6. Sum divisor cordial labeling of fl7 �K1 is shown in Figure 6 as an illustration for

Theorem 2.6.

v0

v1

v1'

v2

v2'

v3

v3'v4

v4'

v5v5'

v6

v6'v7

v7'

u1'

u1

u2u2'

u3

u3'

u4

u4'

u5

u5'

u6' u6

u7

u7'

v0'6

4

12

10

14

1

3

5

11

7

13

2

17

15

19

18

16

20 8

923

22

21

24

27

2526

28

29

30

Figure 6

Definition 2.4 ([4]). The fan Fn is defined as the join of Pn and K1. The vertex corresponding to

K1 is said to be the apex vertex. The fan Fn is shell Sn+1.

Theorem 2.7. Fn �K1 is a sum divisor cordial graph.

Proof. Let v0, v1, v2, . . . , vn be the vertices of the fan Fn, where v0 be apex vertex.

Let v′0, v′1, v′2, . . . , v

′n be the newly added vertices to obtain the graph Fn �K1.

V (Fn �K1) = V (Fn) ∪ {vi, ; 0 ≤ i ≤ n}.10



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E(Fn �K1) = E(Fn) ∪ {viv′i; 0 ≤ i ≤ n}.|V (Fn �K1)| = 2n + 2.

|E(Fn �K1)| = 3n.

We define labeling f : V (Fn �K1)→ {1, 2, 3, . . . , 2n + 2} as follows.

For n ≡ 0, 3(mod4) :

f(vi) =

i + 1 i ≡ 0, 1(mod 4);

i + 2 i ≡ 2(mod 4);

i i ≡ 3(mod 4); 1 ≤ i ≤ n.

f(v0) = 1;

f(v′0) = 2n + 2;

f(v′i) = (n + 1) + i 1 ≤ i ≤ n.

For n ≡ 1(mod4) :

f(vi) =

i + 1 i ≡ 0, 1(mod 4);

i + 2 i ≡ 2(mod 4);

i i ≡ 3(mod 4); 1 ≤ i ≤ n.

f(v0) = 1;

f(v′0) = 2n + 1;

f(v′i) = (n + 1) + i 1 ≤ i ≤ n− 1;

f(v′n) = 2n + 2.

For n ≡ 2(mod4) :

f(vi) =

i + 1 i ≡ 0, 1(mod 4);

i + 2 i ≡ 2(mod 4);

i i ≡ 3(mod 4); 1 ≤ i ≤ n.

f(v0) = 1;

f(v′0) = 2n + 1;

f(v′i) = n + 1 + i 2 ≤ i ≤ n− 1;

f(v′1) = n + 1;

f(v′n) = 2n + 2.



n ≡ 1(mod 4) ef (1) =⌈3n2

⌋, ef (0) =

⌊3n2

⌉

n ≡ 3(mod 4) ef (1) =⌊3n2

⌋, ef (0) =

⌈3n2

⌉

n ≡ 0, 2(mod 4) ef (1) = 3n2 = ef (0)

Thus |ef (0)− ef (1)| ≤ 1.

Hence the graph under consideration admits sum divisor cordial labeling.

Hence Fn �K1 is a sum divisor cordial graph.

Example 2.7. Sum divisor cordial labeling of F8 � K1 is shown in Figure 7 as an illustration for

Theorem 2.7.

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v1 v2 v4 v5

v2'v1' v3' v4' v5'

u1'

u1

v6 v7

v6' v7'

v3 v8

v8'

64 8

1210 14

1

3 5

11

97

13

2

1715

18

16

Figure 7

Definition 2.5 ([4]). The Double fan DFn is defined as the join Pn + 2K1.

Theorem 2.8. DFn �K1 is a sum divisor cordial graph.

Proof. Let u0, v0, v1, v2, . . . , vn be the vertices of the double fan DFn, where u0 and v0 are the vertices

of degree n and v1, v2, . . . , vn are the vertices corresponding to path Pn.

Let u′0, v′0, v′1, v′2, . . . , v

′n be the newly added vertices to obtain the graph DFn �K1.

V (DFn �K1) = V (DFn) ∪ {u′0} ∪ {v′i, ; 0 ≤ i ≤ n}.E(DFn �K1) = E(DFn) ∪ {u0u

′0} ∪ {viv′i; 0 ≤ i ≤ n}.

|V (DFn �K1)| = 2n + 4.

|E(DFn �K1)| = 4n + 1.

We define labeling f : V (DFn �K1)→ {1, 2, 3, . . . , 2n + 4} as follows.

For n ≡ 0, 2(mod4)

f(u0) = 1;

f(u′0) = 2n + 4;

f(v0) = 2;

f(v′0) = 2n + 3;

f(vi) = 2 + i 1 ≤ i ≤ n;

f(v′i) = n + 2 + i 1 ≤ i ≤ n.

For n ≡ 1, 3(mod4)

f(u0) = 1;

f(u′0) = 2n + 2;

f(v0) = 2;

f(v′0) = 2n + 4;

f(vi) = 2 + i 1 ≤ i ≤ n;

f(v′2i−1) = n + (2i− 1) + 3 1 ≤ i ≤ n + 1

2;

f(v′2i) = n + 2i + 1 1 ≤ i ≤ n− 1

2.



n ≡ 1, 3(mod 4) ef (1) =⌈4n+1

2

⌉, ef (0) =

⌊4n+1

2

⌋

n ≡ 0, 2(mod 4) ef (1) =⌊4n+1

2

⌋, ef (0) =

⌈4n+1

2

⌉

12



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Thus |ef (0)− ef (1)| ≤ 1.


Hence DFn �K1 is a sum divisor cordial graph.

Example 2.8. Sum divisor cordial labeling of DF6 �K1 is shown in Figure 8 as an illustration for

Theorem 2.8.

v4 v5

v2'v1' v3' v4' v5'

u1'

u2

u2'

u1

v6

v6'

v2 v3

3 4 5 6 7 8

1

2

15

16

9 10 11 12 13 14

v1

Figure 8

Definition 2.6 ([4]). Let G = (V,E) be a graph. If every edge of graph G is subdivided, then the

resulting graph is called barycentric subdivision of graph G.

In other words, barycentric subdivision is the graph obtained by inserting a vertex of degree 2 into

every edge of original graph.

The barycentric subdivision of any graph G is denoted by S(G).

Theorem 2.9. S(K1,n)�Kn is a sum divisor cordial graph.

Proof. Let v0, v1, v2, . . . , vn, w1, w2, . . . , wn be the vertices of graph S(K1,n), where v0 be apex vertex,

v1, v2, . . . , vn be the vertices of degree 1 and w1, w2, . . . , wn be the vertices of degree 2.

Let v′1, v′2, . . . , v

′n, w′1, w

′2, . . . , w

′n be the newly added vertices to obtain the graph S(K1,n)�K1.

V (S(K1,n)�K1) = V (S(K1,n)) ∪ {v′0, v′i, w′i; 1 ≤ i ≤ n}.E(S(K1,n)�K1) = E(S(K1,n) ∪ {v0v′0, viv′i, wiw

′i; 1 ≤ i ≤ n}.

|V (S(K1,n)�K1)| = 4n + 2.

|E(S(K1,n)�K1)| = 4n + 1.

We define labeling f : V (S(K1,n)�K1)→ {1, 2, 3, . . . , 4n + 2} as follows.

f(v0) = 1;

f(v′0) = 4n + 2;

f(wi) = 2i + 1 1 ≤ i ≤ n;

f(w′i) = 2i 1 ≤ i ≤ n;

f(vi) = (2n + 1) + 2i 1 ≤ i ≤ n;

f(v′i) = 2n + 2i 1 ≤ i ≤ n.


ef (1) =

⌊4n + 1

2

⌋,

ef (0) =

⌈4n + 1

2

⌉.

13



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Thus |ef (0)− ef (1)| ≤ 1.


Hence S(K1,n)�K1 is a sum divisor cordial graph.

Example 2.9. Sum divisor cordial labeling of S(K1,5)�K1,5 is shown in Figure 9 as an illustration

for Theorem 2.9.

u1

v0 v1 v1'

v2

v2'

v3

v3'

v4

v4'v5

v5'

u2

u2'

u1'

u3

u4

u4'

u5

u3'

u5'

v0'

1

3 2

4

7

6

8

9

10

11

12

14

13

15

5

16

17

18

19

20

22

21

Figure 9

Definition 2.7 ([4]). A chord of a cycle Cn is an edge joining two non-adjacent vertices of cycle Cn.

Theorem 2.10. G�K1 is a sum divisor cordial graph, where G is cycle with one chord and chord

forms a triangle with two edges of Cn.

Proof. Let G be the cycle Cn with one chord.

Let v1, v2, . . . , vn be vertices of Cn and e = v2vn be the chord of Cn.

Let v′1, v′2, . . . , v

′n be the newly added vertices to obtain the graph G�K1.

V (G�K1) = V (Cn) ∪ {v′i; 1 ≤ i ≤ n},E(G�K1) = E(Cn) ∪ {viv′i; 1 ≤ i ≤ n}.Hence |V (G�K1)| = 2n and |E(G�K1)| = 2n + 1.

We define labeling f : V (G�K1)→ {1, 2, 3, . . . , 2n} as follows.

f(vi) = 2i− 1 1 ≤ i ≤ n;

f(v′i) = 2i 1 ≤ i ≤ n.


ef (1) =

⌈2n + 1

2

⌉,

ef (0) =

⌊2n + 1

2

⌋.

Thus |ef (0)− ef (1)| ≤ 1.

Hence G�K1 is a sum divisor cordial graph.

Example 2.10. Sum divisor cordial labeling of corona of C6 with one chord and K1 is shown in

Figure 10 as an illustration for Theorem 10.

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v1

v2

v3

v4

v5

v6

v1'

v2'

v3'

v4'

v5'

v6'

1

3

4

6

7

8

5

2

9

10

11

12

Figure 10

Definition 2.8 ([4]). Two chords of a cycle are said to be twin chords if they form a triangle with

an edge of the cycle Cn.

For positive integers n and p with 3 ≤ p ≤ n−2, Cn,p denotes the graph consisting of a cycle Cn with

twin chords with which the edges of Cn form cycles Cp, C3 and Cn+1−p without chords.

Theorem 2.11. Cn,3 �K1 is a sum divisor cordial graph.

Proof. Let v1, v2, . . . , vn be the vertices of Cn, e1 = v2vn and e2 = v3vn be the chords of Cn.

Let v′1, v′2, . . . , v

′n be the newly added vertices to obtain the graph Cn,3 �K1.

V (Cn,3 �K1) = V (Cn,3) ∪ {v′i; 1 ≤ i ≤ n}.E(Cn,3 �K1) = E(Cn,3) ∪ {viv′i; 1 ≤ i ≤ n}.|V (Cn,3 �K1)| = 2n.

|E(Cn,3 �K1)| = 2n + 2.

We define labeling f : V (Cn,3 �K1)→ {1, 2, 3, . . . , 2n} as follows.

For n ≡ 0, 1, 3(mod4) :

f(vi) =

i i ≡ 0, 1(mod 4);

i + 1 i ≡ 2(mod 4);

i− 1 i ≡ 3(mod 4); 1 ≤ i ≤ n.

f(v′i) = n + 1 1 ≤ i ≤ n.

For n ≡ 2(mod4) :

f(vi) =

i i ≡ 0, 1(mod 4);

i + 1 i ≡ 2(mod 4);

i− 1 i ≡ 3(mod 4); 1 ≤ i ≤ n.

f(v′i) = n + i 3 ≤ i ≤ n;

f(v′1) = n;

f(v′2) = n + 2.


ef (1) = n + 1 = ef (0)

15



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Thus |ef (0)− ef (1)| ≤ 1.

Hence Cn,3 �K1 is a sum divisor cordial graph.

Example 2.11. Sum divisor cordial labeling of C7,3 �K1 is shown in Figure 11 as an illustration

for Theorem 11.

v1

v1'

v2

v2'

v3 v3'

v4

v4'

v5

v5'

v6v6'

v7

v7' 1

3

2

45

7

6

8

9

10

1112

14

13

Figure 11

Definition 2.9 ([4]). A cycle with triangle is a cycle with three chords which by themselves form a

triangle.

For positive integers p, q, r and n ≥ 6 with p+ q + r + 3 = n, Cn(p, q, r) denotes a cycle with triangle

whose edges form the edges of cycles Cp+2, Cq+2 and Cr+2 without chords.

Theorem 2.12. Cn(1, 1, n− 5)�K1 is a sum divisor cordial graph.

Proof. Let v1, v2, . . . , vn be the vertices of Cn.

Let e1 = u1u3, e2 = u3un−1 and e3 = u1un−1 be chords of Cn which by themselves form a triangle.

Let v′1, v′2, . . . , v

′n be the newly added vertices to obtain the graph Cn(1, 1, n− 5)�K1.

V (Cn(1, 1, n− 5)�K1) = V (Cn) ∪ {v′i; 1 ≤ i ≤ n},E(Cn(1, 1, n− 5)�K1) = E(Cn(1, 1, n− 5)) ∪ {viv′i; 1 ≤ i ≤ n.}Hence |V (Cn(1, 1, n− 5)�K1)| = 2n and |E(Cn(1, 1, n− 5)�K1)| = 2n + 3.

We define labeling f : V (Cn(1, 1, n− 5)�K1)→ {1, 2, 3, . . . , 2n} as follows.

For n ≡ 0, 1, 3(mod4) :

f(vi) =

i i ≡ 0, 1(mod 4);

i + 1 i ≡ 2(mod 4);

i− 1 i ≡ 3(mod 4); 1 ≤ i ≤ n.

f(v′i) = n + i 1 ≤ i ≤ n.

For n ≡ 2(mod4) :

f(vi) =

i i ≡ 0, 1(mod 4);

i + 1 i ≡ 2(mod 4);

i− 1 i ≡ 3(mod 4); 1 ≤ i ≤ n.

f(v′i) = n + i 3 ≤ i ≤ n;

f(v′1) = n;

f(v′2) = n + 2.

16



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ef (1) =

⌊2n + 3

2

⌋,

ef (0) =

⌈2n + 3

2

⌉.

Thus |ef (0)− ef (1)| ≤ 1.

Hence Cn(1, 1, n− 5)�K1 is a sum divisor cordial graph.

Example 2.12. Sum divisor cordial labeling of C8(1, 1, 3)�K1 is shown in Figure 12 as an illustration

for Theorem 12.

v1

v1'

v2

v2'

v3 v3'

v4

v4'v5

v5'

v6

v6'

v7

v8

v7'

1

3

2

4

5

7

6

8

9

10

11

12

13

16

15

14

Figure 12

3 Concluding Remarks

The sum divisor cordial labeling is an invariant of divisor cordial labeling by considering codomain

as finite set of numbers. It is interesting to see that if two graphs are sum divisor cordial then their

corona is sum divisor cordial or not. We have investigated twelve sum divisor cordial graphs in context

of corona of graphs.

References

[1] D. G. Adalja and G. V. Ghodasara, Some New Sum Divisor Cordial Graphs, International

Journal of Applied Graph Theory, Vol. 2, No. 1, 2018, pp. 19 - 33.

[2] D. M. Burton, Elementary Number Theory, Brown Publishers, Second Edition, (1990).

[3] R. Frucht and F. Harary, On Corona of Two Graphs, Aequationes Mathematicae, Vol. 4, No. 3,

1970. DOI: 10.1007/BF01817769

[4] J. A. Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics,

20(2017), # DS6, pp. 1 - 432.

[5] G. V. Ghodasara and D. G. Adalja, Divisor Cordial Labeling in Context of Ring Sum of Graphs,

International Journal of Mathematics and Soft Computing, Vol. 7, No. 1, 2017, pp. 23 - 31.

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[6] J. Gross and J. Yellen, Graph Theory and Its Applications, CRC Press, (2004).

[7] A. Lourdusamy and F. Patrick, J. Shiama, Sum Divisor Cordial Graphs, Proyecciones Journal

of Mathematics, Vol. 35, No. 1, 2016, pp. 119 - 136.

[8] A. Lourdusamy and F. Patrick, Sum Divisor Cordial Labeling For Star And Ladder Related

Graphs, Proyecciones Journal of Mathematics, Vol. 35, No. 4, 2016, pp. 437 - 455.

[9] A. Rosa, On certain valuations of the vertices of theory of graphs, (Internat.Symposium, Rome,

July 1966) Gordon and Breach, N. Y. and Dunod Paris (1967), pp. 349 - 355.

[10] R. Varatharajan and S. Navanaeethakrishnan and K. Nagarajan, Divisor Cordial Graphs, Inter-

national J. Math. Combin., Vol. 4, 2011, pp. 15 - 25.

[11] V. Yegnanaryanan and P. Vaidhyanathan, Some Interesting Applications of Graph Labellings,

J. Math. Comput. Sci., Vol. 2, No.5, 2012, pp.1522 - 1531.

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