international journal of mathematics and scientific...

INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: 2231-5330), VOL. 2, NO. 1, 2012 3

3-Equitable Labeling for Some Star and BistarRelated Graphs

S.K. Vaidya and N.H. Shah

Abstract—In this paper we prove that the splitting graphs ofK1,n and Bn,n are 3-equitable graphs. We also show that theshadow graph of Bn,n is a 3-equitable graph. Further we provethat square graph of Bn,n is 3-equitable for n ≡ 0(mod 3)and n ≡ 1(mod 3) and not 3-equitable for n ≡ 2(mod 3).

Index Terms—3-equitable labeling, star, bistar, splitting graph,shadow graph, square graph.

MSC 2010 Codes - 05C78.

I. INTRODUCTION

IN this paper we consider simple, finite, connected andundirected graph G = (V (G), E(G)) with order p andsize q. For all standard terminology and notations we followWest [9]. We will give brief summary of definitions whichare useful for the present investigations.

Definition 1.1 : If the vertices of the graph are assigned valuessubject to certain condition(s) then is known as graph labeling.

A detailed study on applications of graph labeling isreported in Bloom and Golomb [3]. According to Beinekeand Hegde [2] graph labeling serves as a frontier betweennumber theory and structure of graphs. For an extensivesurvey on graph labeling and bibliographic references werefer to Gallian [4].

Definition 1.2 : Let G = (V (G), E(G)) be a graph. Amapping f : V (G) →{0,1,2} is called ternary vertex labelingof G and f(v) is called the label of the vertex v of G under f .

For an edge e = uv, the induced edge labelingf∗ : E(G) → {0, 1, 2} is given by f∗(e) = |f(u)−f(v)|. Letvf (0), vf (1) and vf (2) be the number of vertices of G havinglabels 0,1 and 2 respectively under f and let ef (0),ef (1)and ef (2) be the number of edges having labels 0,1 and 2respectively under f∗.

Definition 1.3 : A ternary vertex labeling of a graph Gis called a 3-equitable labeling if |vf (i) − vf (j)| ≤ 1 and|ef (i) − ef (j)| ≤ 1 for all 0 ≤ i, j ≤ 2. A graph G is3-equitable if it admits 3-equitable labeling.

Dr. S. K. Vaidya is a proffesor at the Department of Mathematics, SaurashtraUniversity, Rajkot, Gujarat - 360005, INDIA(e-mail: [email protected])

N. H. Shah is a lecturer at Government Polytechnic, Rajkot, Gujarat -360003, INDIA(e-mail: [email protected])

The concept of 3-equitable labeling was introduced byCahit [1] and he proved that an Eulerian graph with numberof edges congruent to 3(mod 6) is not 3-equitable. In thesame paper he proved that Cn is 3-equitable if and onlyif n 6≡ 3(mod 6) and all caterpillars are 3-equitable. Cahit[1] claimed to prove that Wn is 3-equitable if and only ifn 6≡ 3(mod 6) but Youssef [8] proved that Wn is 3-equitablefor all n ≥ 4. The 3-equitable labeling in the context ofvertex duplication is discussed by Vaidya et al. [5] whilesame authors in [6] have investigated 3-equitable labling forsome shell related graphs. Vaidya et al. [7] have discussed 3-equitablity of graphs in the context of some graph operationsand proved that the shadow and middle graph of cycle Cn,path Pn are 3-equitable.

Generally there are three types of problems that can beconsidered in this area.

1) How 3-equitability is affected under various graph op-erations?

2) Construct new families of 3-equitable graph by investi-gating suitable labeling.

3) Given a graph theoretic property P, characterize the classof graphs with property P that are 3-equitable.

The problems of second type are largely discussed whilethe problems of first and third types are rarely discussed andthey are of great importance also. The present work is aimedto discuss the problems of first kind.

Definition 1.4 : The splitting graph of a graph G is obtainedby adding to each vertex v a new vertex v′ such that v′

is adjacent to every vertex that is adjacent to v in G, i.e.N(v) = N(v′). The resultant graph is denoted by S′(G).

Definition 1.5: The shadow graph D2(G) of a connectedgraph G is constructed by taking two copies of G say G′

and G′′. Join each vertex u′ in G′ to the neighbours of thecorresponding vertex u′′ in G′′.

Definition 1.6: For a simple connected graph G the square ofgraph G is denoted by G2 and defined as the graph with thesame vertex set as of G and two vertices are adjacent in G2

if they are at a distance 1 or 2 apart in G.


II. MAIN RESULTSTheorem 2.1 : S′(K1,n) is 3-equitable graph.Proof : Let v1, v2, v3, . . . , vn be the pendant vertices andv be the apex vertex of K1,n and u, u1, u2, u3, . . . , un areadded vertices corresponding to v, v1, v2, v3, . . . , vn to obtainS′(K1,n). Let G be the graph S′(K1,n) then |V (G)| = 2n+2and |E(G)| = 3n. To define f : V (G) → {0, 1, 2} we considerfollowing three cases.Case 1: n ≡ 0(mod 3)

f(v) = 2,f(u) = 0,

f(vi) = 0; 1 ≤ i ≤ n3+ 1

f(vn

3 +1+i

)= 1; 1 ≤ i ≤ n

3− 1

f(v 2n

3 +i

)= 2; 1 ≤ i ≤ n

3

f(ui) = 0; 1 ≤ i ≤ n3− 1

f(un

3 −1+i)= 1; 1 ≤ i ≤ n

3+ 2

f(u 2n

3 +1+i

)= 2; 1 ≤ i ≤ n

3− 1

In view of the above labeling patten we have

vf (0) = vf (1) =2n

3+ 1 = vf (2) + 1

ef (0) = ef (1) = ef (2) = n

Case 2: n ≡ 1(mod 3)

Since n ≡ 1(mod 3), n = 3k + 1 some k ∈ N.f(v) = 2,f(u) = 0,f(vi) = 0; 1 ≤ i ≤ k + 1f (vk+1+i) = 1; 1 ≤ i ≤ k − 1f (v2k+i) = 2; 1 ≤ i ≤ k + 1f(ui) = 0; 1 ≤ i ≤ k − 1f (uk−1+i) = 1; 1 ≤ i ≤ k + 3f (u2k+2+i) = 2; 1 ≤ i ≤ k − 1


vf (0) = vf (2) =2n+ 1

3= vf (1)− 1

ef (0) = ef (1) = ef (2) = n

Case 3: n ≡ 2(mod 3)

Since n ≡ 2(mod 3), n = 3k + 2 some k ∈ N.f(v) = 2,f(u) = 0,f(vi) = 0; 1 ≤ i ≤ k + 1f (vk+1+i) = 1; 1 ≤ i ≤ kf (v2k+1+i) = 2; 1 ≤ i ≤ k + 1f(ui) = 0; 1 ≤ i ≤ kf (uk+i) = 1; 1 ≤ i ≤ k + 2f (u2k+2+i) = 2; 1 ≤ i ≤ k


vf (0) = vf (2) = vf (1) =

⌊2(n+ 1)

3

⌋

ef (0) = ef (1) = ef (2) = n

Thus in each case we have |vf (i) − vf (j)| ≤ 1 and|ef (i)− ef (j)| ≤ 1, for all 0 ≤ i, j ≤ 2.Hence S′(K1,n) is a 3-equitable graph.

Illustration 2.2 : 3-equitable labeling of the graph S′(K1,7)is shown in Fig. 1.

v

u4

u

v4

v1

v7

v6

v2 v

3 v5

u1

u2

u3 u5

u6

u7

10

2

1

11

1

2

22

0

2

0

1

0

0

Figure 1

Theorem 2.3 : S′(Bn,n) is 3-equitable graph.Proof : Consider Bn,n with vertex set {u, v, ui, vi, 1 ≤ i ≤ n}where ui, vi are pendant vertices. In order to obtain S′(Bn,n),add u′, v′, u′i, v

′i vertices corresponding to u, v, ui, vi where,

1 ≤ i ≤ n. If G = S′(Bn,n) then |V (G)| = 4(n + 1) and|E(G)| = 6n+3. To define f : V (G) → {0, 1, 2} we considerfollowing four cases.Case 1: n = 2, 5The graphs S′(B2,2) and S′(B5,5) are to be dealt separatelyand their 3-equitable labeling is shown is Fig. 2. and Fig. 3.

u

u2

u1

2 0

2 2

v1 v2

2 0

1 1

v

v1' v

2'u

1' u

2'

u' v '

10

0 1

Figure 2

u

u1

u2 u

3

u4

u5

2

22

0

0

12

22

2v1

v2 v

3

v4

v5

2

10

0

0

01

11

1

v

v1'

v2' v3

'v4'

v5'u

1'

u2' u3

'u4'

u5'

u' v '

10

0 1

Figure 3


Case 2: n ≡ 0(mod 3)Since n ≡ 0(mod 3), n = 3k some k ∈ N.

f(u) = 0,f(u′) = 0,f(v) = 1,f(v′) = 1,f(ui) = 2; 1 ≤ i ≤ nf (u′1) = 2;f(u′1+i

)= 1; 1 ≤ i ≤ k

f(u′k+1+i

)= 0; 1 ≤ i ≤ n− k − 1

f(vi) = 0; 1 ≤ i ≤ 2(k − 1)f(v2k−2+i) = 1; 1 ≤ i ≤ n− 2k + 2f (v′i) = 2; 1 ≤ i ≤ kf(v′k+i

)= 1; 1 ≤ i ≤ 2k − 2

f (v′i) = 0; i = n, n− 1In view of the above labeling patten we have

vf (0) = vf (2) = 4k + 1 = vf (1)− 1ef (0) = ef (1) = ef (2) = 2n+ 1

Case 3: n ≡ 1(mod 3)Since n ≡ 1(mod 3), n = 3k + 1 some k ∈ N.


)= 1; 1 ≤ i ≤ k

f(u′k+1+i

)= 0; 1 ≤ i ≤ 2k

f(vi) = 1; 1 ≤ i ≤ k + 2f(vk+2+i) = 0; 1 ≤ i ≤ 2k − 1f (v′i) = 2; 1 ≤ i ≤ kf(v′k+i

)= 1; 1 ≤ i ≤ 2k − 1

f (v′i) = 0; i = n, n− 1In view of the above labeling patten we have

vf (0) = vf (1) = 4k + 3 = vf (2) + 1

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

Case 4: n ≡ 2(mod 3)Since n ≡ 2(mod 3), n = 3k + 2 some k ∈ N− {1}.


)= 1; 1 ≤ i ≤ k + 1

f(u′k+2+i

)= 0; 1 ≤ i ≤ 2k

f(vi) = 1; 1 ≤ i ≤ k + 4f(vk+4+i) = 0; 1 ≤ i ≤ 2k − 2f (v′i) = 2; 1 ≤ i ≤ k + 1f(v′k+1+i

)= 1; 1 ≤ i ≤ 2k − 3

f (v′i) = 0; n− 3 ≤ i ≤ n


vf (0) = vf (1) = 4(k + 1) = vf (2)

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

Thus in each case we have |vf (i) − vf (j)| ≤ 1 and|ef (i)− ef (j)| ≤ 1, for all 0 ≤ i, j ≤ 2.Hence S′(Bn,n) is a 3-equitable graph.

Illustration 2.4 : 3-equitable labeling of the graph S′(B6,6)is shown in Fig. 4.

u

u1

u3 u

4

u5

u6

1

10

0

0

2

22

22

v1

v2 v

3 v4

v5

2

21

1

0

10

01

1

v

v1'

v2' v3

'v4'

v5'u

2'

u3' u4

'u5'

u6'

u' v '

10

0 1

2

u2

u1'

2

v6'

0

1

v6

Figure 4

Theorem 2.5 : D2(Bn,n) is 3-equitable graph.Proof : Consider two copies of Bn,n. Let{u, v, ui, vi, 1 ≤ i ≤ n} and {u′, v′, u′i,v′i, 1 ≤ i ≤ n}be the corresponding vertex sets of each copy of Bn,n. LetG be the graph D2(Bn,n) then |V (G)| = 4(n + 1) and|E(G)| = 4(2n + 1). To define f : V (G) → {0, 1, 2} weconsider following three cases.Case 1: n ≡ 0(mod 3)Since n ≡ 0(mod 3), n = 3k some k ∈ N.

f(u) = 0,f(u′) = 2,f(v) = 0,f(v′) = 2,f(ui) = 0; 1 ≤ i ≤ 2k + 1f(u2k+1+i) = 1; 1 ≤ i ≤ k − 1f (u′i) = 1; 1 ≤ i ≤ 2(k − 1)f(u′2(k−1)+i

)= 2; 1 ≤ i ≤ k + 2

f(vi) = 0; 1 ≤ i ≤ 2k − 1f(v2k−1+i) = 1; 1 ≤ i ≤ k + 1f (v′i) = 1; 1 ≤ i ≤ 3f(v′3+i

)= 2; 1 ≤ i ≤ n− 3

vf (1) = vf (2) =4(n+ 1)− 1

3= vf (0)− 1

ef (0) = ef (2) =8n+ 3

3= ef (1)− 1



f(u) = 0,f(u′) = 2,f(v) = 0,f(v′) = 2,f(ui) = 0; 1 ≤ i ≤ nf (u′i) = 2; 1 ≤ i ≤ nf(vi) = 0; 1 ≤ i ≤ kf(vk+i) = 1; 1 ≤ i ≤ n− kf (v′i) = 1; 1 ≤ i ≤ 2k + 1f(v′2k+1+i

)= 2; 1 ≤ i ≤ k

vf (0) = vf (2) =4(n+ 1) + 1

3= vf (1) + 1

ef (0) = ef (2) = ef (1) =8n+ 4

3

Case 3: n ≡ 2(mod 3)Since n ≡ 2(mod 3), n = 3k + 2 some k ∈ N ∪ {0}.

f(u) = 0,f(u′) = 0,f(v) = 0,f(v′) = 1,f(ui) = 0; 1 ≤ i ≤ k + 1f(uk+1+i) = 1; 1 ≤ i ≤ kf(u2k+1+i) = 2; 1 ≤ i ≤ k + 1f (u′i) = 2; 1 ≤ i ≤ nf(vi) = 0; 1 ≤ i ≤ n− 2f(vn−1) = 1;f(vn) = 2;f (v′i) = 1; 1 ≤ i ≤ n

vf (0) = vf (1) = vf (2) =4(n+ 1)

3

ef (0) = ef (2) =8n+ 4 + 1

3= ef (1) + 1

Thus in each case we have |vf (i) − vf (j)| ≤ 1 and|ef (i)− ef (j)| ≤ 1, for all 0 ≤ i, j ≤ 2.Hence D2(Bn,n) is a 3-equitable graph.

Illustration 2.6 : 3-equitable labeling of the graph D2(B5,5)is shown in Fig. 5.

u1

u2

3 u4

u5

u1'

u2'

u3'

u4'

u5'

v1

v2

3 v4

v5

v1'

v2' v

4'

v5'

u

u'

v

v '

0

0

01

2

2 0

01

2

1

11

1

1

1

0

0

2

22

2

2

0

v3'

Figure 5

Theorem 2.7 : B2n,n is 3-equitable graph for n ≡ 0(mod 3)and n ≡ 1(mod 3).

Proof : Consider Bn,n with vertex set {u, v, ui, vi, 1 ≤ i ≤ n}where ui, vi are pendant vertices. Let G be the graph B2n,nthen |V (G)| = 2n + 2 and |E(G)| = 4n + 1. To definef : V (G) → {0, 1, 2}, we consider following two cases.Case 1: n ≡ 0(mod 3)Since n ≡ 0(mod 3), n = 3k some k ∈ N.

f(u) = 2,f(v) = 0,f(ui) = 1; 1 ≤ i ≤ kf(uk+i) = 2; 1 ≤ i ≤ 2kf(vi) = 0; 1 ≤ i ≤ 2kf(v2k+i) = 1; 1 ≤ i ≤ k

vf (0) = vf (2) =2n+ 3

3= vf (1) + 1

ef (0) = ef (1) =4n

3= ef (2)− 1


f(u) = 2,f(v) = 0,f(ui) = 1; 1 ≤ i ≤ kf(uk+i) = 2; 1 ≤ i ≤ 2k + 1f(vi) = 0; 1 ≤ i ≤ 2kf(v2k+i) = 1; 1 ≤ i ≤ k + 1

vf (0) = vf (1) =2n+ 1

3= vf (2)− 1

ef (1) = ef (2) =4n+ 2

3= ef (0) + 1

Thus in both the cases we have |vf (i) − vf (j)| ≤ 1 and|ef (i)− ef (j)| ≤ 1, for all 0 ≤ i, j ≤ 2.Hence B2n,n is a 3-equitable graph for n ≡ 0(mod 3) andn ≡ 1(mod 3).

Illustration 2.8 : 3-equitable labeling of the graph B27,7 isshown in Fig. 6.

v1

v2

v3

v4

v6

v

1

1

22

1

v5

v7

u

u1

u2

u3 u

4

u5

u6

u7

2

0

1

2

2

1

00

0

0

2

Figure 6


Theorem 2.9 : B2n,n is not a 3-equitable graph for n ≡ 2(mod3).Proof : Let G be the graph B2n,n then |V (G)| = 2n + 2and |E(G)| = 4n + 1. Here n ≡ 2(mod 3) thereforen = 3k1 + 2 for some k1 ∈ N. Hence |V (G)| = 3k and|E(G)| = 6k − 3 where k = 2k1 + 2 . So if B2n,n is3-equitable then we must have vf (0) = vf (1) = vf (2) = kand ef (0) = ef (1) = ef (2) = 2k − 1.

In B2n,n, note that each ui and vi (1 ≤ i ≤ n) are adjacentto u and v both moreover u and v are adjacent vertices. It isobvious that any edge will have label 1 if it is incident to onevertex with label 1. Following Table 1. shows all possibleassignments of vertex label. From the Table 1 (column 6)we can observe that the edge condition violates in all thepossible assignments. Hence B2n,n is not a 3-equitable graphfor n ≡ 2(mod 3).

III. CONCLUDING REMARKS

The graphs K1,n and Bn,n are 3-equitable being caterpillarswhile we show that the splitting graphs of K1,n and Bn,nalso admit 3-equitable labeling. Thus 3-equitability remainsinvariant for the splitting graphs of K1,n and Bn,n. It is alsoinvariant for shadow graph of Bn,n. Moreover we prove thatthe square graph of Bn,n is 3-equitable for n ≡ 0(mod 3)and n ≡ 1(mod 3) while not 3-equitable for n ≡ 2(mod 3).To investigate similar results for other graph families and forvarious graph operations is a potential area of research.

REFERENCES[1] I. Cahit, “On cordial and 3-equitable labelings of graphs”, Util. Math.,

vol. 37, pp. 189-198, 1990.[2] L. W. Beineke and S. M. Hegde,“Strongly multiplicative graphs”, Discuss.

Math. Graph Theory, vol. 21, pp. 63-75, 2001.[3] G. S. Bloom and S. W. Golomb, “Applications of numbered undirected

graphs”, Proc. of IEEE, vol. 65, pp. 562-570, 1977.[4] J. A. Gallian, “A dynamic survey of graph labeling”, The Electronic

Journal of Combinatorics, vol. 18, #DS6, 2011.Available online: http://www.combinatorics.org/Surveys/ds6.pdf

[5] S. K. Vaidya, N. A. Dani, K. K. Kanani and P. L. Vihol, “Some wheelrelated 3-equitable graphs in the context of vertex duplication”, Advancesand Applications in Discrete Mathematics, vol. 4, no. 1, pp. 71-85, 2009.Available online: http://www.pphmj.com/journals/aadm.htm

[6] S. K. Vaidya, N. A. Dani, K. K. Kanani and P. L. Vihol, “Cordial and3-equitable labeling for some shell related graphs”, J. Sci. Res., vol. 1,no. 3, pp. 438-449, 2009.Available online: http://www.banglajol.info/index.php/JSR/article/view/2227/2617

[7] S. K. Vaidya, P. L. Vihol and C. M. Barasara, “3-equitable graphs in thecontext of some graph operations”, Journal of Applied Computer Scienceand Mathematics, vol 11, no 5, pp. 69-75, 2011.Available online: http://jacs.usv.ro/index.php

[8] M. Z. Youssef, “A necessary condition on k-equitable labelings”, Util.Math., vol. 64, pp. 193-195, 2003.

[9] D. B. West, “Introduction to Graph Theory”, Prentice-Hall of India, 2001.


TABLE I

u v ui’s and vi’sef (1)

vertex label vertex label v′f (0) v′f (1) v

′f (2)

0 0 k − 2 k k 2k 6= 2k − 1

0 1 k − 1 k − 1 k 3k − 1 6= 2k − 1

0 2 k − 1 k k − 1 2k 6= 2k − 1

1 1 k k − 2 k 4k 6= 2k − 1

1 2 k k − 1 k − 1 3k − 1 6= 2k − 1

2 2 k k k − 2 2k 6= 2k − 1

Where v′f (j) = number of vertices having label j for ui and vi where 1 ≤ i ≤ n and 0 ≤ j ≤ 2 .

International Journal of Mathematics and Soft ComputingVol.3, No.1 (2013), 61 - 68. ISSN Print : 2249 - 3328

ISSN Online: 2319 - 5215

Graceful and odd graceful labeling of some graphs

S K VaidyaSaurashtra University, Rajkot - 360005,

Gujarat, INDIA.E-mail: [email protected]

N H ShahGovernment Polytechnic, Rajkot - 360003,

Gujarat, INDIA.E-mail: [email protected]

Abstract

In this paper, we prove that the square graph of bistarBn,n, the splitting graph ofBn,n and

the splitting graph of starK1,n are graceful graphs. We also prove that the splitting graph and the

shadow graph of bistarBn,n admit odd graceful labeling.

Keywords: Graceful labeling, odd graceful labeling, shadow graph, splitting graph, square graph.

AMS Subject Classification(2010):05C78.

1 Introduction

We begin with simple, finite, connected and undirected graphG = (V (G), E(G)) with |V (G)| = pand |E(G)| = q. For standard terminology and notation we follow Gross and Yellen [5]. We pro-vide a brief summary of definitions and other information which serve as prerequisites for the present

investigation.

Definition 1.1. If the vertices are assigned values subject to certain condition(s) then it is known as

graph labeling.

Graph labelings is an active area of research in graph theory which has rigorous applications in

coding theory, communication networks, optimal circuits layouts and graph decomposition problems.

According to Beineke and Hegde [1] graph labeling serves as a frontier between number theory and the

structure of graphs. For a dynamic survey of various graph labeling problems along with an extensive

bibliography we refer to Gallian [2].

Definition 1.2. A function f is called graceful labelingof a graph G if f : V (G) → {0, 1, 2, . . . , q} isinjective and the induced function f∗ : E(G) → {1, 2, . . . , q} defined as f∗(e = uv) = |f(u)− f(v)|is bijective. The graph which admits graceful labeling is called a graceful graph.

Rosa [9] initially called this labeling aβ − valuation and later Golomb [4] named it asgracefullabelingwhich is now the popular term. Several infinite families of graceful and non-graceful graphs

have been studied. The famous Ringel-Kotzig tree conjecture [8] and many illustrious work on graceful

61

62 S K Vaidya and N H Shah

graphs brought a tide of labeling scheme having graceful theme. Vaidya et al. [11] discussed grace-

fulness of union of two path graphs with grid graph and complete bipartite graph. Kaneria et al. [6]

discussed gracefulness of some classes of disconnected graphs. Vaidya and Lekha [12] investigated

graceful labeling of some cycle related graphs. Some variants of graceful labeling were also introduced

in recent past, such as edge graceful labeling, Fibonacci graceful labeling, odd graceful labeling and the

like.

Definition 1.3. A function f is called odd graceful labeling of a graph G if f : V (G) → {0, 1, 2, . . . ,2q−1} is injective and the induced function f∗ : E(G) → {1, 3, . . . , 2q − 1} defined as f∗(e = uv) =|f(u)−f(v)| is bijective. The graph which admits odd graceful labeling is called an odd graceful graph.

The concept of odd graceful labeling was introduced by Gnanajothi [3]. It is possible to decompose

the graphKn,n with suitable odd graceful labeling of tree T of ordern. The splitting graphs of path

Pn and even cycleCn are proved to be odd graceful by Sekar [10] while ladders and graphs obtained

from them by subdividing each step exactly once are shown odd-graceful by Kathiresan [7]. Vaidya and

Lekha [13–15] proved many results on odd graceful labeling.

The following three types of problems are considered generally in the area of graph labeling.

1. How particular labeling is affected under various graph operations;

2. Investigation of new families of graphs which admit particular graph labeling;

3. Given a graph theoretic property P, characterizing the class/classes of graphs with property P that

admit particular graph labeling.

From the literature survey, it is clear that the problems of second type are largely studied than the

problems of first and third types. The present work is aimed to discuss some problems of the first kind

in the context of graceful and odd graceful labeling.

Definition 1.4. For a graph G the splitting graphS′ of G is obtained by adding a new vertex v′ corre-

sponding to each vertex v of G such that N(v) = N(v′).

Definition 1.5. The shadow graphD2(G) of a connected graph G is constructed by taking two copiesof G say G′ and G′′. Join each vertex u′ in G′ to the neighbours of the corresponding vertex v′ in G′′.

Definition 1.6. For a simple connected graph G the square of graphG is denoted by G2 and defined as

the graph with the same vertex set as of G and two vertices are adjacent in G2 if they are at a distance 1

or 2 apart in G.

2 Results on graceful labeling

Theorem 2.1.B2n,n is a graceful graph.

Proof. ConsiderBn,n with the vertex set{u, v, ui, vi, 1 ≤ i ≤ n} whereui, vi are the pendant vertices.Let G be the graphB2n,n then|V (G)| = 2n + 2 and|E(G)| = 4n + 1. We define the vertex labelingf : V (G) → {0, 1, 2, . . . , 4n + 1} as follows.

Graceful and odd graceful labeling of some graphs 63

f(v) = 0,f(u) = 4n + 1,

f(vi) = i; 1 ≤ i ≤ nf(ui) = f(vn) + i; 1 ≤ i ≤ n.

The vertex functionf defined above induces a bijective edge functionf∗ : E(G) → {1, 2, 3, . . . ,4n + 1}. Thusf is graceful labeling ofG = B2n,n. Hence,B2n,n is a graceful graph.

Illustration 2.2. Graceful labeling of the graph B27,7 is shown in Figure 1.

v 1

v2

v3v4

v6

v

5

6

1210

9

v5

v7

u

u 1

u2u3 u4

u5

u6

u 7

29

0

8

13

14

7

43

2

1

11

Figure 1: Graceful labeling ofB27,7.

Theorem 2.3.S′(Bn,n) is a graceful graph.

Proof. ConsiderBn,n with the vertex set{u, v, ui, vi, 1 ≤ i ≤ n} whereui, vi are the pendant vertices.In order to obtainS′(Bn,n), addu′, v′, u′i, v

′i vertices corresponding tou, v, ui, vi where,1 ≤ i ≤ n.

If G = S′(Bn,n) then |V (G)| = 4(n + 1) and |E(G)| = 3(2n + 1). To define the vertex labelingf : V (G) → {0, 1, 2, 3, . . . , 6n + 3}, we consider the following two cases.Case 1:n is even.

f(u) = 2n− 1,f(v) = 0,f(u′) = 2n,f(v′) = 1,f(u′1) = 4n,

f(u′1+i) = n− 1 + i; 1 ≤ i ≤ n− 1

f(ui) = 1 + 2i; 1 ≤ i ≤n− 2

2f

(un

2−1+i

)= 5n + 4− 2i; 1 ≤ i ≤ n + 2

2

f(v′1+i) = 6n + 3− i; 0 ≤ i ≤ n− 1f(v1) = f(v′n)− 1,f(v1+i) = f(v1)− 2i; 1 ≤ i ≤ n− 1

Case 2:n is odd.


f(u) = 2n− 1,f(v) = 0,f(u′) = 2n,f(v′) = 1,f(u′1) = 2,

f(u′1+i) = n− 1 + i; 1 ≤ i ≤ n− 1

f(ui) = 2(i + 1); 1 ≤ i ≤n− 3

2f

(un−3

2+i

)= 5n + 4− 2i; 1 ≤ i ≤ n + 3

2f(v′i) = 6n + 4− i; 1 ≤ i ≤ nf(v1) = f(v′n)− 1,f(v1+i) = f(v1)− 2i; 1 ≤ i ≤ n− 1

The vertex functionf defined above induces a bijective edge functionf∗ : E(G) → {1, 2, 3, . . . ,6n + 3}. Thus,f is a graceful labeling ofG = S′(Bn,n). Hence,S′(Bn,n) is a graceful graph.

Illustration 2.4. Graceful labeling of the graph S′(B5,5) is shown in Figure 2.

u

u 1u 2 u 3

u4u 5

2

56

7

8

214

2725

23 v1

v2 v3v4

v5

29

3031

32

33

2028

2624

22

v

v1'v2'

v3' v4'

v5'u 1'u2'

u 3' u 4'

u 5'

u' v '110

9 0

Figure 2: Graceful labeling ofS′(B5,5).

Illustration 2.5. Graceful labeling of the graph S′(B6,6) is shown in Figure 3.

u

u1u 3 u 4

u 5u 6

6

78

9

10

265

3230

28 v1

v2 v3 v4

v5

39

3837

36

35

2533

3129

27

v

v1'v2'

v3' v4'

v5'u 2'u 3'

u 4' u 5'

u 6'

u' v '112

11 0

24

u2

u 1'

3

v6'34

23

v6

Figure 3: Graceful labeling ofS′(B6,6).

Theorem 2.6.S′(K1,n) is a graceful graph.

Proof. Let v1, v2, v3, . . . , vn be the pendant vertices andv be the apex vertex ofK1,n andu, u1, u2, u3,


. . . , un be added vertices corresponding tov, v1, v2, v3, . . . , vn to obtain S′(K1,n). Let G be thegraphS′(K1,n) then |V (G)| = 2n + 2 and |E(G)| = 3n. We define vertex labelingf : V (G) →{0, 1, 2, . . . , 3n} as follows.f(u) = 1, f(v) = 0,f(vi) = 2i; 1 ≤ i ≤ nf(ui) = f(vn) + i; 1 ≤ i ≤ n

The vertex functionf defined above induces a bijective edge functionf∗ : E(G) → {1, 2, 3, . . . , 3n}.Thus,f is a graceful labeling ofG = S′(K1,n). Hence,S′(K1,n) is a graceful graph.

Illustration 2.7. Graceful labeling of the graph S′(K1,7) is shown in Figure 4.

v

u 4

u

v4

v1 v7v6v2 v3 v5

u 1u 2

u 3 u 5u 6

u 7

86

21

20

1918

17

1412104

0

16

2

1

15

Figure 4: Graceful labeling of the graphS′(K1,7).

3 Results on Odd Graceful labeling

Theorem 3.1.S′(Bn,n) is an odd graceful graph.

Proof. ConsiderBn,n with the vertex set{u, v, ui, vi, 1 ≤ i ≤ n} whereui, vi are the pendant vertices.In order to obtainS′(Bn,n), addu′, v′, u′i, v

′i vertices corresponding tou, v, ui, vi where,1 ≤ i ≤ n.

If G = S′(Bn,n) then |V (G)| = 4(n + 1) and |E(G)| = 3(2n + 1). We define vertex labelingf : V (G) → {0, 1, 2, 3, . . . , 12n + 5} as follows.f(u) = 0,f(v) = 3,f(u′) = 2,f(v′) = 5,

f(ui) = 5 + 4i; 1 ≤ i ≤ nf(u′i) = 12n + 7− 2i; 1 ≤ i ≤ nf(v′1) = f(u

′n) + 1,

f(v′1+i) = f(v′1)− 2i; 1 ≤ i ≤ n− 1

f(v1) = f(v′n)− 2,f(v1+i) = f(v1)− 4i; 1 ≤ i ≤ n− 1


The vertex functionf defined above induces a bijective edge functionf∗ : E(G) → {1, 3, . . . ,12n + 5}. Thus,f is an odd graceful labeling ofG = S′(Bn,n). Hence,S′(Bn,n) is an odd gracefulgraph.

Illustration 3.2. Odd graceful labeling of the graph S′(B6,6) is shown in Figure 5.

u

u1u 3 u 4

u 5u 6

75

7371

69

67

2913

1721

25 v1

v2 v3 v4

v5

68

6664

62

60

4056

5248

44

v

v1'v2'

v3' v4'

v5'u 2'u 3'

u 4' u 5'

u 6'

u' v '52

0 3

77

u2

u 1'

9

v6'58

36

v6

Figure 5: Odd graceful labeling ofS′(B6,6).

Theorem 3.3.D2(Bn,n) is an odd graceful graph.

Proof. Consider two copies ofBn,n. Let {u, v, ui, vi : 1 ≤ i ≤ n} and{u′, v′, u′i, v′i : 1 ≤ i ≤ n}be the corresponding vertex sets of each copy ofBn,n. Let G be the graphD2(Bn,n). Then|V (G)| =4(n + 1) and|E(G)| = 4(2n + 1). We define vertex labelingf : V (G) → {0, 1, 2, 3, . . . , 16n + 7} asfollows.

f(u) = 2,f(v) = 7,f(u′) = 0,f(v′) = 3,f(ui) = 16n + 11− 4i; 1 ≤ i ≤ nf(u′i) = f(un)− 4i; 1 ≤ i ≤ nf(v1+2i) = 16 + 8i; 0 ≤ i ≤

⌈n2

⌉− 1

f(v2+2i) = 18 + 8i; 0 ≤ i ≤⌊n

2

⌋− 1

f(v′1+2i) = f(vn−1) + 8(i + 1); 0 ≤ i ≤⌈n

2

⌉− 1

f(v′2+2i) = f(vn) + 8(i + 1); 0 ≤ i ≤⌊n

2

⌋− 1

The vertex functionf defined above induces a bijective edge functionf∗ : E(G) → {1, 3, . . . ,16n + 7}. Thus,f is an odd graceful labeling forG = D2(Bn,n). Hence,D2(Bn,n) is an odd gracefulgraph.


Illustration 3.4. Odd graceful labeling of the graph D2(Bn,n) is shown in Figure 6.

u 1

u 2 3 u 4u 5

u 1'

u 2' u 3'u 4'

u 5'

v1

v2 3 v4v5

v1'

v2' v4'

v5'

u

u'

v

v '

2

87

8379

75

71 16

2426

32

50

4842

40

34

3

7

0

67

6359

55

51

18

v 3'

Figure 6: Odd graceful labeling of the graphD2(Bn,n).

4 Concluding Remarks

Vaidya and Lekha [13] have proved that splitting graph of starK1,n is an odd graceful graph while

we prove that it is also a graceful graph. We proved that splitting graph of bistarBn,n is both graceful

and odd graceful. MoreoverB2n,n is a graceful graph but not an odd graceful graph as it contains odd

cycles.

Rosa [9] proved thatK1,n andBn,n are graceful graphs but we prove that the splitting graphs of star

K1,n and bistarBn,n are also graceful graphs. Thus, gracefulness remains invariant for the splitting

graphs ofK1,n andBn,n. It is also invariant for square graph ofBn,n. Moreover odd gracefulness is

invariant for the splitting graph and the shadow graph ofBn,n.

References

[1] L. W. Beineke and S. M. Hegde,Strongly multiplicative graphs, Discuss. Math. Graph Theory ,

21(2001), 63-75.

[2] J. A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics,

18(2011)#DS6.

[3] R. B. Gnanajothi,Topics in graph theory, Ph. D Thesis, Madurai Kamaraj University, Madurai,

(1991).

[4] S. W. Golomb,How to number a graph in Graph Theory and ComputingR C Read, ed., Academic

Press, New York (1972), 23-37.

[5] J. Gross and J. Yellen,Graph theory and its applications, CRC Press, (1999).

[6] V. J. Kaneria, S. K. Vaidya, G. V. Ghodasara and S. Srivastav,Some classes of disconnected grace-

ful graphs, Proceedings of the First International Conference on Emerging Technologies and Ap-

plications in Engineering, Technology and Sciences, (2008), 1050 - 1056.


[7] K. M. Kathiresan,Some classes of odd graceful graphs, ANJAC Journal of Sciences, 7 (1) (2008),

5-10.

[8] G. Ringel, Problem 25, Theory of graphs and its applications, Proc. of Symposium Smolenice

1963, Prague, (1964), 162.

[9] A. Rosa,On certain valuations of the vertices of a graph, Theory of graphs, International Sympo-

sium, Rome, July (1966), Gordon and Breach, New York and Dunod Paris (1967), 349-355.

[10] C. Sekar,Studies in graph theory, Ph. D Thesis, Madurai Kamaraj University, Madurai, (2002).

[11] S. K. Vaidya, V. J. Kaneria, S. Srivastav and N. A. Dani,Gracefulness of union of two path graphs

with grid graph and complete bipartite graph, Proceedings of the First International Conference

on Emerging Technologies and Applications in Engineering, Technology and Sciences,(2008), 616

- 619.

[12] S. K. Vaidya and Lekha Bijukumar,Some new graceful graphs, International Journal of Mathe-

matics and Soft Computing, 1(1)(2011), 37-45.

[13] S. K. Vaidya and Lekha Bijukumar,Odd graceful labeling of some new graphs, Modern Applied

Science, 4(10)(2010), 65-70.

[14] S. K. Vaidya and Lekha Bijukumar,Some new odd graceful graphs, Advances and Applications in

Discrete Mathematics, 6(2)(2010), 101-108.

[15] S. K. Vaidya and Lekha Bijukumar,New families of odd graceful graphs, International Journal of

Open Problems in Computer Science and Mathematics, 3(5)(2010), 166-171.

Annals of Pure and Applied Mathematics Vol. 3, No. 1, 2013, 67-77 ISSN: 2279-087X (P), 2279-0888(online) Published on 31 May 2013 www.researchmathsci.org

67

Annals of

Some Star and Bistar Related Divisor Cordial Graphs

S. K. Vaidya 1 and N. H. Shah2

1Department of Mathematics, Saurashtra University, Rajkot – 360005, Gujarat, India

E-mail: [email protected]

2Government Polytechnic, Rajkot-360003, Gujarat, India E-mail: [email protected]

Received 19 May 2013; accepted 27 May 2013

Abstract. A divisor cordial labeling of a graph G with vertex set V is a bijection f from V to {1, 2,... | |}V such that an edge uv is assigned the label 1 if either ( ) | ( )f u f v or

( ) | ( )f v f u and the label 0 if ( ) ( )f u f v , then number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph with a divisor cordial labeling is called a divisor cordial graph. In this paper we prove that splitting graphs of star 1,nK and bistar ,n nB are divisor cordial graphs. Moreover we show that degree splitting graph of ,n nB , shadow graph of ,n nB and square graph of ,n nB admit divisor cordial labeling.. Keywords: Divisor cordial labeling, star, bistar. AMS Mathematics Subject Classification (2010): 05C78 1. Introduction We begin with simple, finite, connected and undirected graph ( ( ), ( ))G V G E G= with p vertices and q edges. For standard terminology and notations related to graph theory

we refer to Gross and Yellen [1] while for number theory we refer to Burton [2] . We will provide brief summary of definitions and other information which are necessary for the present investigations. Definition 1.1. If the vertices are assigned values subject to certain condition(s) then it is known as graph labeling. Any graph labeling will have the following three common characteristics: 1. A set of numbers from which vertex labels are chosen; 2. A rule that assigns a value to each edge;

S. K. Vaidya and N. H. Shah

68

3. A condition that this value has to satisfy. According to Beineke and Hegde [3] graph labeling serves as a frontier between

number theory and structure of graphs. Graph labelings have many applications within mathematics as well as to several areas of computer science and communication networks. According to Graham and Sloane [4] the harmonious labelings are closely related to problems in error correcting codes while odd harmonious labeling is useful to solve undetermined equations as described by Liang and Bai [5]. The optimal linear arrangement concern to wiring network problems in electrical engineering and placement problems in production engineering can be formalised as a graph labeling problem as stated by Yegnanaryanan and Vaidhyanathan [6]. For a dynamic survey of various graph labeling problems along with an extensive bibliography we refer to Gallian [7]. Definition 1.2. A mapping : ( ) {0,1}f V G → is called binary vertex labeling of G and

( )f v is called the label of the vertex v of G under f . Notation 1.3. If for an edge e uv= , the induced edge labeling * : ( ) {0,1}f E G → is given by * ( ) | ( ) ( ) |f e f u f v= − . Then

( ) number of vertices of having label under

( ) number of vertices of having label under *f

f

v i G i f

e i G i f

=

=

Definition 1.4. A binary vertex labeling f of a graph G is called a cordial labeling if | (0) (1) | 1f fv v− ≤ and | (0) (1) | 1f fe e− ≤ . A graph G is cordial if it admits cordial labeling.

The concept of cordial labeling was introduced by Cahit [8]. This concept is explored by many researchers like Andar et al. [9,10], Vaidya and Dani [11] and Nasreen [12]. Motivated through the concept of cordial labeling Babujee and Shobana [13] introduced the concepts of cordial languages and cordial numbers. Some labeling schemes are also introduced with minor variations in cordial theme. Product cordial labeling, total product cordial labeling and prime cordial labeling are among mention a few. The present work is focused on divisor cordial labeling. Definition 1.5. A prime cordial labeling of a graph G with vertex set ( )V G is a bijection

: ( ) {1, 2,3, , | ( ) |}f V G V G→ … and the induced function * : ( ) {0,1}f E G → is defined by * ( ) 1, if gcd( ( ), ( )) 1;

0, otherwise.f e uv f u f v= = =

=

which satisfies the condition | (0) (1) | 1f fe e− ≤ . A graph which admits prime cordial labeling is called a prime cordial graph.

The concept of prime cordial labeling was introduced by Sundaram et al. [14] and in the same paper they have investigated several results on prime cordial labeling. Vaidya and Vihol [15,16] as well as Vaidya and Shah [17,18] have proved many results on prime cordial labeling.


69

Motivated by the concept of prime cordial labeling, Varatharajan et al. [19] introduced a new concept called divisor cordial labeling by combining the divisibility concept in Number theory and Cordial labeling concept in Graph labeling. This is defined as follows. Definition 1.6. Let ( ( ), ( ))G V G E G= be a simple graph and : {1,2,... | ( ) |}f V G→ be a bijection. For each edge uv , assign the label 1 if either ( ) | ( )f u f v or ( ) | ( )f v f u and the label 0 if ( ) ( )f u f v . f is called a divisor cordial labeling if | (0) (1) | 1f fe e− ≤ . A graph with a divisor cordial labeling is called a divisor cordial graph.

In the same paper [19] they have proved that path, cycle, wheel, star, 2,nK and

3,nK are divisor cordial graphs while nK is not divisor cordial for 7n ≥ . Same authors in [20] have discussed divisor cordial labeling of full binary tree as well as some star related graphs.

It is important to note that prime cordial labeling and divisor cordial labeling are two independent concepts. A graph may possess one or both of these properties or neither as exhibited below.

i) Pn (n≥ 6) is both prime cordial as proved in [14] and divisor cordial as proved in [19].

ii) C3 is not prime cordial as proved in [14], but it is divisor cordial as proved in [19].

iii) We found that a 7-regular graph with 12 vertices admits prime cordial labeling, but does not admit divisor cordial labeling.

iv) Complete graph K7 is not a prime cordial as stated in Gallian [7] and not divisor cordial as proved in [19].

Generally there are three types of problems that can be considered in the area of

graph labeling. 1. How a particular labeling is affected under various graph operations; 2. To investigate new graph families which admit particular graph labeling; 3. Given a graph theoretic property P, characterize the class/classes of graphs with property P that admit particular graph labeling.

The problems of second type are largely discussed while the problems of first and third types are not so often but they are of great importance. The present work is aimed to discuss the problems of first kind in the context of divisor cordial labeling. Definition 1.7. For a graph G the splitting graph ( )S G′ of a graph G is obtained by adding a new vertex 'v corresponding to each vertex v of G such that ( ) ( ')N v N v= . Definition 1.8. [21] Let ( ( ), ( ))G V G E G= be a graph with 1 2 3 iV S S S S T= ∪ ∪ ∪… ∪ where each iS is a set of vertices having at least two vertices of the same degree and

iT V S= ∪ . The degree splitting graph of G denoted by ( )DS G is obtained from G by adding vertices 1 2 3, , , , tw w w w… and joining to each vertex of iS for 1 i t≤ ≤ .


70

Definition 1.9. The shadow graph 2 ( )D G of a connected graph G is constructed by taking two copies of G say G′ and ''G . Join each vertex u′ in G′ to the neighbours of the corresponding vertex v′ in G′′ . Definition 1.10. For a simple connected graph G the square of graph G is denoted by

2G and defined as the graph with the same vertex set as of G and two vertices are adjacent in 2G if they are at a distance 1 or 2 apart inG . 2. Main Results Theorem 2.1. 1,( )nS K′ is a divisor cordial graph. Proof : Let 1 2 3, , , , nv v v v… be the pendant vertices and v be the apex vertex of 1,nK and

1 2 3, , , , , nu u u u u… are added vertices corresponding to 1 2 3, , , , , nv v v v v… to obtain 1,( )nS K′ . Let G be the graph 1,( )nS K′ then | ( ) | 2 2V G n= + and | ( ) | 3E G n= . To define : ( ) {1, 2, , 2 2}f V G n→ … + we consider following three cases. Case 1: n=2 to 8 For n=2, f(v)=4, f(u)=1 , f(v1)=3, f(v2)=2, and, f(u1)=5, f(u2)=6. Then (0) 3 (1)f fe e= = . For n=3, f(v)=3, f(u)=2, f(v1)=5, f(v2)=6, f(v3)=7 and f(u1)=1, f(u2)=9, f(u3)=4. Then (0) 5, (1) 4f fe e= = . For n=4, f(v)=3, f(u)=2, f(v1)=5, f(v2)=6, f(v3)=8, f(v4)=10 and f(u1)=1, f(u2)=4, f(u3)=7, f(u4)=9. Then (0) 6 (1)f fe e= = . For n=5 , f(v)=3, f(u)=2, f(v1)=5, f(v2)=6, f(v3)=8, f(v4)=10, f(v5)=12 and f(u1)=1, f(u2)=4, f(u3)=7, f(u4)=9, f(u5)=11. Then (0) 7, (1) 8f fe e= = . For n=6 , f(v)=3, f(u)=2, f(v1)=5, f(v2)=6, f(v3)=8, f(v4)=10, f(v5)=12, f(v6)=14 and f(u1)=1, f(u2)=4, f(u3)=7, f(u4)=9, f(u5)=11, f(u6)=13. Then (0) 9 (1)f fe e= = . For n=7 , f(v)=3, f(u)=2, f(v1)=5, f(v2)=6, f(v3)=8, f(v4)=10, f(v5)=12, f(v6)=14 , f(v7)=16 and f(u1)=1, f(u2)=4, f(u3)=7, f(u4)=9, f(u5)=11, f(u6)=13, f(u7)=15. Then (0) 10, (1) 11f fe e= = . For n=8, f(v)=3, f(u)=2, f(v1)=1, f(v2)=6, f(v3)=12, f(v4)=5, f(v5)=7, f(v6)=11 , f(v7)=13, f(v8)=17 and f(u1)=4, f(u2)=8, f(u3)=10, f(u4)=14, f(u5)=16, f(u6)=18, f(u7)=9, f(u7)=15. Then (0) 12 (1)f fe e= = . Now for the remaining two cases let,

( )1 2 2 3, 1 , 2 .3 3 2

n n ns m s t n s

+ += = − − = − + +⎢ ⎥ ⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎢ ⎥

1 2 1 3 4 31, , , .x s t x n x x n s m t x n x= + + = − = − + − = − Case 2: n=9, 10, 11, 12 (t=0)


71

( ) 2, ( ) 3,f v f u= =

1

1

( ) 1,( ) 6 ; 1i

f vf v i i s+

== ≤ ≤

( )

( )

( )

1

1

21 2

22 2

1 2

2 2

5 6 ; 02

7 6 ; 02

( ) 4 6 ; 02

( ) 8 6 ; 02

9 6( 1); 1

x i

x i

i

i

n s i

xf v i i

xf v i i

n sf u i i

n sf u i i

f u i i m

+ +

+ +

+

+

− +

⎡ ⎤= + ≤ < ⎢ ⎥⎢ ⎥⎢ ⎥= + ≤ < ⎢ ⎥⎣ ⎦

−⎡ ⎤= + ≤ < ⎢ ⎥⎢ ⎥−⎢ ⎥= + ≤ < ⎢ ⎥⎣ ⎦

= + − ≤ ≤

For the vertices 3 31 2

( ), ( ), , ( )x x nf u f u f u+ + … assign distinct remaining odd numbers. This assigns all the vertex labels for case 2. Case 3: 13 ( 1)n t≥ ≥

( ) 2, ( ) 3,f v f u= =

1

1

( ) 1,( ) 6 ; 1i

f vf v i i s+

== ≤ ≤

( )

( )

( )

( )

1

1

1

21 2

22 2

1 2

2 2

1

9 6( 1); 1

5 6 ; 02

7 6 ; 02

( ) 4 6 ; 02

( ) 8 6 ; 02

( ) 6 ; 1

s i

x i

x i

i

i

n s i s t

f v i i txf v i i

xf v i i

n sf u i i

n sf u i i

f u f v i i m t

+ +

+ +

+ +

+

+

− + + +

= + − ≤ ≤

⎡ ⎤= + ≤ < ⎢ ⎥⎢ ⎥⎢ ⎥= + ≤ < ⎢ ⎥⎣ ⎦

−⎡ ⎤= + ≤ < ⎢ ⎥⎢ ⎥−⎢ ⎥= + ≤ < ⎢ ⎥⎣ ⎦

= + ≤ ≤ −

For the vertices 3 31 2

( ), ( ), , ( )x x nf u f u f u+ + … assign distinct remaining odd numbers. This assigns all the vertex labels for case 3. In view of the above labeling pattern,

1( ) | ( ), , ( ) | ( )n sf v f u f v f u −… and 1 1( ) | ( ), ( ) | ( )f v f v f v f u . Moreover

2 3 1( ) | ( ), ( ) | ( ), , ( ) | ( )sf v f v f v f v f v f v +… , 2 3( ) | ( ), ( ) | ( ), ,f u f v f u f v … 1( ) | ( )sf u f v + and 2 3 1( ) | ( ), ( ) | ( ), , ( ) | ( )s s s tf u f v f u f v f u f v+ + + +… .


72

Hence,3

(1) 2 2 ( 2)2fn

e n s s s t n s n S= − + + + + = + + + − + +⎡ ⎤⎢ ⎥⎢ ⎥.

Therefore, in last two case 3

(1)2fn

e = ⎡ ⎤⎢ ⎥⎢ ⎥and

3(0)

2fn

e = ⎢ ⎥⎢ ⎥⎣ ⎦.

Thus, in all the cases we have | (0) (1) | 1f fe e− ≤ .

Hence, 1,( )nS K′ is a divisor cordial graph. Example 2.2. Let 1,13( )G S K′= , | ( ) | 28V G = and | ( ) | 39E G = . In accordance with

Theorem 2.1 we have 1 2 3 44, 4, 1, 6, 7, 12, 1s m t x x x x= = = = = = = and using the labeling pattern described in case 2. The corresponding divisor cordial labeling is shown in Figure 1. It is easy to visualize that (0) 19 and (1) 20f fe e= = .

Figure 1: Divisor cordial labeling of 1,13( )G S K′=

Theorem 2.3. ,( )n nS B′ is a divisor cordial graph. Proof: Consider ,n nB with vertex set { , , , ,1 }i iu v u v i n≤ ≤ where ,i iu v are pendant vertices. In order to obtain ,( )n nS B′ , add , , ,i iu v u v′ ′ ′ ′ vertices corresponding to

, , ,, i iu v u v where 1 i n≤ ≤ . If ,( )n nG S B′= then | ( ) | 4( 1)V G n= + and | ( ) | 6 3E G n= + . We define vertex labeling : ( ) {1, 2, , 4( 1)}f V G n→ … + as follows. Let 1p be the highest prime number 4( 1)n< + .

( )

1

( ) 2, ( ) 1,( ) 4, ( ) ,( ) 6 2( 1); 1

( ) 2 ; 1i

ni

f u f uf v f v pf u i i nf u f u i i n

′= =

′= =

= + − ≤ ≤

′ = + ≤ ≤

For the vertices 1 2, , , nv v v… and 1 2, , , nv v v′ ′ ′… we assign distinct odd numbers (except 1p ).


73

In view of the above labeling pattern we have, (0) 3 1, (1) 3 2f fe n e n= + = + . Thus, | (0) (1) | 1f fe e− ≤ . Hence, ,( )n nS B′ is a divisor cordial graph. Illustration 2.4. Divisor cordial labeling of the graph 6,6( )S B′ is shown in Figure 2.

Figure 2: Divisor cordial labeling of the graph 6,6( )S B′

Theorem 2.5. ,( )n nDS B is a divisor cordial graph. Proof: Consider ,n nB with ,( ) { , , , :1 }n n i iV B u v u v i n= ≤ ≤ , where ,i iu v are pendant vertices. Here , 1 2( )n nV B V V= ∪ , where 1 { , :1 }i iV u v i n= ≤ ≤ and 2 { , }V u v= . Now in order to obtain ,( )n nDS B from G , we add 1 2,w w corresponding to 1 2,V V . Then

,| ( ( ) | 2 4n nV DS B n= + and

, 2 2 1 1( ( )) { , , } { , , , :1 }n n i i i iE DS B uv uw vw uu vv w u w v i n= ∪ ≤ ≤ so ,| ( ( ) | 4 3n nE DS B n= + . We define vertex labeling ,: ( ( )) {1,2, , 2 4}n nf V DS B n→ … + as follows.

( ) 4,f u = ( ) 2 3,f v n= +

1( ) 1,f w = 2( ) 2,f w = ( ) 3 2( 1); 1( ) 6 2( 1); 1

i

i

f u i i nf v i i n

= + − ≤ ≤

= + − ≤ ≤

In view of the above defined labeling patten we have, (0) 2 2, (1) 2 1f fe n e n= + = + .

Thus, | (0) (1) | 1f fe e− ≤ .

Hence, ,( )n nDS B is a divisor cordial graph. Illustration 2.6. Divisor cordial labeling of the graph 5,5( )DS B is shown in Figure 3.


74

Figure 3: Divisor cordial labeling of the graph 5,5( )DS B

Theorem 2.7. 2 ,( )n nD B is a divisor cordial graph. Proof: Consider two copies of ,n nB . Let { , , , ,1 }i iu v u v i n≤ ≤ and ,{ , , ,1 }i iu v u v i n′ ′ ′ ′ ≤ ≤ be the corresponding vertex sets of each copy of ,n nB . Let G be the graph 2 ,( )n nD B then | ( ) | 4( 1)V G n= + and| ( ) | 4(2 1)E G n= + . We define vertex labeling f: V(G) →{1,2,…, 4(n+1)} as follows. Let 1p be the highest prime number and 2p be the second highest prime number such that 2 1 4( 1)p p n< < + .

( )

1 2

1

( ) 2, ( ) 1,( ) , ( ) ,( ) 6 2( 1); 1

( ) 2 ; 1( ) 4,

i

i n

f u f uf v p f v pf u i i nf u f u i i nf v

′= =′= =

= + − ≤ ≤′ = + ≤ ≤=

For the vertices 2 3, , , nv v v… and 1 2, , , nv v v′ ′ ′… we assign distinct odd numbers (except

1p and 2p ). In view of the above defined labeling pattern we have, (0) 4 2 (1)f fe n e= + = . Thus, | (0) (1) | 1f fe e− ≤ . Hence, 2 ,( )n nD B is a divisor cordial graph. Illustration 2.8. Divisor cordial labeling of graph 2 5,5( )D B is shown in Figure 4.


75

Figure 4: Divisor cordial labeling of graph 2 5,5( )D B

Theorem 2.9. 2,n nB is a divisor cordial graph. Proof: Consider ,n nB with vertex set { , , , ,1 }i iu v u v i n≤ ≤ where ,i iu v are pendant

vertices. Let G be the graph 2,n nB then | ( ) | 2 2V G n= + and| ( ) | 4 1E G n= + . We define vertex labeling : ( ) {1, 2, , 2 2}f V G n→ … + as follows. Let 1p be the highest prime number


76

3. Concluding Remarks As all the graphs are not divisor cordial graphs it is very interesting and challenging as well to investigate divisor cordial labeling for the graph or graph families which admit divisor cordial labeling. Here we have contributed some new results by investigating divisor cordial labeling for some star and bistar related graphs.

Varatharajan et al. [19] have proved that 1,nK and ,n nB are divisor cordial graphs while we prove that the splitting graphs of star 1,nK and bistar ,n nB are also divisor cordial graphs. Thus divisor cordiality remains invariant for the splitting graphs of 1,nK and ,n nB . It is also invariant for degree splitting graph of ,n nB , shadow graph of ,n nB and square graph of ,n nB .

Acknowledgement: The authors are highly thankful to the anonymous referee for kind comments and constructive suggestions.

REFERENCES

1. J. Gross and J. Yellen, Graph Theory and its applications, CRC Press, (1999). 2. D. M. Burton, Elementary Number Theory, Brown Publishers, Second Edition,

(1990). 3. L. W. Beineke and S. M. Hegde, Strongly multiplicative graphs, Discuss. Math.

Graph Theory, 21 (2001) 63-75. 4. R. L. Graham and N. J. A. Sloane, On additive bases and harmonious graphs, SIAM

J. Alg. Disc. Meth., 1 (4) (1980) 382-404. 5. Z. Liang and Z. Bai, On the odd harmonious graphs with applications, J. Appl. Math.

Comput., 29 (2009) 105-116. 6. V. Yegnanaryanan and P. Vaidhyanathan, Some interesting applications of graph

labellings, J. Math. Comput. Sci., 2 (5) (2012) 1522-1531. 7. J. A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of

Combinatorics, 19, (2012), # DS6. Available: http://www.combinatorics.org/Surveys/ds6.pdf

8. I. Cahit, Cordial Graphs, A weaker version of graceful and harmonious Graphs, Ars Combinatoria, 23 (1987), 201-207.

9. M. Andar, S. Boxwala and N. B. Limaye, Cordial labelings of some wheel related graphs, J. Combin. Math. Combin. Comput., 41 (2002) 203-208.

10. M. Andar, S. Boxwala and N. B. Limaye, A note on cordial labeling of multiple shells, Trends Math., (2002) 77-80.

11. S. K. Vaidya and N. A. Dani, Some new star related graphs and their cordial as well as 3-equitable labeling, Journal of Sciences, 1 (1) (2010) 111-114.

12. Nasreen Khan, Cordial labelling of cactus graphs, Advanced Modeling and Optimization, 15(1) (2013) 85-101.

13. J. B. Babujee and L. Shobana, Cordial Languages and Cordial Numbers, Journal of Applied Computer Science & Mathematics, 13 (6) (2012) 9-12.


77

14. M. Sundaram, R. Ponraj and S. Somasundram, Prime Cordial Labeling of graphs, J. Indian Acad. Math., 27 (2) (2005) 373-390.

15. S. K. Vaidya and P. L. Vihol, Prime cordial labeling for some graphs, Modern Applied Science, 4 (8) (2010) 119-126. Available: http://ccsenet.org/journal/index.php/mas/

16. S. K. Vaidya and P. L. Vihol, Prime cordial labeling for some cycle related graphs, Int. J. of Open Problems in Computer Science and Mathematics, 3 (5) (2010) 223-232.Available: http://www.ijopcm.org/Vol/10/IJOPCM%28vol.3.5.24.D.10%29.pdf

17. S. K. Vaidya and N. H. Shah, Some new families of prime cordial graphs, J. of Mathematics Research, 3 (4) (2011) 21-30. Available: http://www.ccsenet.org/journal/index.php/jmr/article/view/9650/8944

18. S. K. Vaidya and N. H. Shah, Prime cordial labeling of some graphs, Open Journal of Discrete Mathematics, 2 (2012) 11-16. doi:10.4236/ojdm.2012.21003 Available: http://www.scirp.org/journal/PaperInformation.aspx?paperID=17154

19. R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan, Divisor cordial graphs, International J. Math. Combin., 4 (2011) 15-25.

20. R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan, Special classes of divisor cordial graphs, International Mathematical Forum, 7 (35) (2012) 1737-1749.

21. P. Selvaraju, P. Balaganesan, J. Renuka and V. Balaj, Degree splitting graph on graceful, felicitous and elegant labeling, International J. Math. Combin., 2 (2012) 96-102.

Malaya Journal of Matematik 4(1)(2013) 148–156

Prime cordial labeling of some wheel related graphs

S. K. Vaidyaa,∗and N. H. Shahb

aDepartment of Mathematics, Saurashtra University, Rajkot - 360005, Gujarat, India.

bDepartment of Mathematics, Government Polytechnic, Rajkot - 360003, Gujarat, India.

Abstract

A prime cordial labeling of a graph G with the vertex set V (G) is a bijection f : V (G) → {1, 2, 3, . . . , |V (G)|}such that each edge uv is assigned the label 1 if gcd(f(u), f(v)) = 1 and 0 if gcd(f(u), f(v)) > 1, then thenumber of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph whichadmits prime cordial labeling is called prime cordial graph. In this paper we prove that the gear graph Gnadmits prime cordial labeling for n ≥ 4. We also show that the helm Hn for every n, the closed helm CHn (forn ≥ 5) and the flower graph Fln (for n ≥ 4) are prime cordial graphs.

Keywords: Prime cordial labeling, gear graph, helm, closed helm, flower graph.

2010 MSC: 05C78. c©2012 MJM. All rights reserved.

1 Introduction

We begin with simple, finite, connected and undirected graph G = (V (G), E(G)) with p vertices and qedges. For standard terminology and notations we follow Gross and Yellen [5]. We will provide brief summaryof definitions and other information which are necessary for the present investigations.

Definition 1.1. If the vertices are assigned values subject to certain condition(s) then it is known as graphlabeling.Any graph labeling will have following three common characteristics:1. a set of numbers from which vertex labels are chosen;2. a rule that assigns a value to each edge;3. a condition that this value has to satisfy.

According to Beineke and Hegde [1] graph labeling serves as a frontier between number theory and structureof graphs. Graph labelings have many applications within mathematics as well as to several areas of computerscience and communication networks. According to Graham and Sloane [4] the harmonious labellings areclosely related to problems in error correcting codes while odd harmonious labeling is useful to solve undeter-mined equations as described by Liang and Bai [6]. The optimal linear arrangement concern to wiring networkproblems in electrical engineering and placement problems in production engineering can be formalised as agraph labeling problem as stated by Yegnanaryanan and Vaidhyanathan [13]. The watershed transform is animportant morphological tool used for image segmentation. An improved algorithm using Graceful labeling forwatershed image segmentation is also proposed by Sridevi et al.[7]. For a dynamic survey on various graphlabeling problems along with an extensive bibliography we refer to Gallian [3].

∗Corresponding author.

E-mail addresses: [email protected] (S. K. Vaidya) and [email protected] (N. H. Shah).

S. K. Vaidya et al. / Prime cordial labeling ... 149

Definition 1.2. A mapping f : V (G) → {0, 1} is called binary vertex labeling of G and f(v) is called the labelof the vertex v of G under f .

Definition 1.3. If for an edge e = uv, the induced edge labeling f∗ : E(G) → {0, 1} is given by f∗(e) =|f(u)− f(v)|. Then

vf (i) = number of vertices of G having label i under fef (i) = number of edges of G having label i under f∗

}where i = 0 or 1

Definition 1.4. A binary vertex labeling f of a graph G is called a cordial labeling if |vf (0)− vf (1)| ≤ 1 and|ef (0)− ef (1)| ≤ 1. A graph G is cordial if it admits cordial labeling.

The concept of cordial labeling was introduced by Cahit [2]. Some labeling schemes are also introduced withminor variations in cordial theme. Product cordial labeling, total product cordial labeling and prime cordiallabeling are among mention a few. The present work is focused on prime cordial labeling.

Definition 1.5. A prime cordial labeling of a graph G with vertex set V (G) is a bijection f : V (G) →{1, 2, 3, . . . , |V (G)|} and the induced function f∗ : E(G) → {0, 1} is defined byf∗(e = uv) = 1, if gcd(f(u), f(v)) = 1;

= 0, otherwise.satisfies the condition |ef (0)− ef (1)| ≤ 1. A graph which admits prime cordial labeling is called a prime cordialgraph.

The concept of prime cordial labeling was introduced by Sundaram et al.[8] and in the same paper theyhave investigated several results on prime cordial labeling. Vaidya and Vihol [9] as well as Vaidya and Shah [12]have discussed prime cordial labeling in the context of some graph operations. Prime cordial labeling for somecycle related graphs have been discussed by Vaidya and Vihol in [10]. Vaidya and Shah [11] have investigatedmany results on prime cordial labeling. Same authors in [12] have proved that the wheel graph Wn admitsprime cordial labeling for n ≥ 8. The present work is aimed to investigate some new results on prime cordiallabeling for some wheel related graphs.

Definition 1.6. The wheel Wn is defined to be the join K1 + Cn. The vertex corresponding to K1 is known asapex and vertices corresponding to cycle are known as rim vertices while the edges corresponding to cycle areknown as rim edges. We continue to recognize apex of wheel as the apex of respective graphs corresponding todefinitions 1.6 to 1.9.

Definition 1.7. The gear graph Gn is obtained from the wheel by subdividing each of its rim edge.

Definition 1.8. The helm Hn is the graph obtained from a wheel Wn by attaching a pendant edge to each rimvertex. It contains three types of vertices: an apex of degree n, n vertices of degree 4 and n pendant vertices.

Definition 1.9. The closed helm CHn is the graph obtained from a helm Hn by joining each pendant vertexto form a cycle. It contains three types of vertices: an apex of degree n, n vertices of degree 4 and n verticesdegree 3.

Definition 1.10. The flower Fln is the graph obtained from a helm Hn by joining each pendant vertex to theapex of the helm. It contains three types of vertices: an apex of degree 2n, n vertices of degree 4 and n verticesof degree 2.

2 Main Results

Theorem 2.1. Gear graph Gn is a prime cordial graph for n ≥ 4.

Proof. Let Wn be the wheel with apex vertex v and rim vertices v1, v2, . . . , vn. To obtain the gear graph Gnsubdivide each rim edge of wheel by the vertices u1, u2, . . . , un. Where each ui is added between vi and vi+1for i = 1, 2, . . . , n− 1 and un is added between v1 and vn. Then |V (Gn)| = 2n+1 and |E(Gn)| = 3n. To definef : V (G) → {1, 2, 3, . . . , 2n + 1}, we consider following four cases.

150 S. K. Vaidya et al. / Prime cordial labeling ...

Case 1: n = 3In G3 to satisfy the edge condition for prime cordial labeling it is essential to label four edges with label 0 andfive edges with label 1 out of nine edges. But all the possible assignments of vertex labels will give rise to 0labels for at most three edges and 1 labels for at least six edges. That is, |ef (0)− ef (1)| = 3 > 1. Hence, G3 isnot prime cordial graph.Case 2: n = 4 to 9, 11, 14, 19For n = 4, f(v) = 6, f(v1) = 3, f(v2) = 9, f(v3) = 4, f(v4) = 8 and f(u1) = 1, f(u2) = 7, f(u3) = 2, f(u4) = 5.Then ef (0) = 6 = ef (1).For n = 5, f(v) = 6, f(v1) = 9, f(v2) = 5, f(v3) = 4, f(v4) = 8, f(v5) = 3 and f(u1) = 7, f(u2) = 10, f(u3) =2, f(u4) = 1, f(u5) = 11. Then ef (0) = 8, ef (1) = 7.For n = 6, f(v) = 6, f(v1) = 9, f(v2) = 8, f(v3) = 4, f(v4) = 11, f(v5) = 1, f(v6) = 10 and f(u1) = 12, f(u2) =2, f(u3) = 13, f(u4) = 5, f(u5) = 7, f(u6) = 3. Then ef (0) = 9 = ef (1).For n = 7, f(v) = 2, f(v1) = 7, f(v2) = 4, f(v3) = 6, f(v4) = 12, f(v5) = 8, f(v6) = 10, f(v7) = 14 andf(u1) = 5, f(u2) = 3, f(u3) = 9, f(u4) = 15, f(u5) = 1, f(u6) = 11, f(u7) = 13. Then ef (0) = 10, ef (1) = 11.For n = 8, f(v) = 2, f(v1) = 1, f(v2) = 4, f(v3) = 6, f(v4) = 12, f(v5) = 8, f(v6) = 10, f(v7) = 14, f(v8) = 16and f(u1) = 7, f(u2) = 3, f(u3) = 9, f(u4) = 15, f(u5) = 5, f(u6) = 11, f(u7) = 13, f(u8) = 17. Thenef (0) = 12 = ef (1).For n = 9, f(v) = 2, f(v1) = 4, f(v2) = 5, f(v3) = 6, f(v4) = 12, f(v5) = 18, f(v6) = 8, f(v7) = 10, f(v8) =14, f(v9) = 16 and f(u1) = 7, f(u2) = 3, f(u3) = 9, f(u4) = 15, f(u5) = 1, f(u6) = 11, f(u7) = 13, f(u8) =17, f(u9) = 19. Then ef (0) = 13, ef (1) = 14.For n = 11, f(v) = 2, f(v1) = 4, f(v2) = 5, f(v3) = 6, f(v4) = 12, f(v5) = 18, f(v6) = 8, f(v7) = 10, f(v8) =14, f(v9) = 16, f(v10) = 20, f(v11) = 22 and f(u1) = 7, f(u2) = 3, f(u3) = 9, f(u4) = 15, f(u5) = 21, f(u6) =11, f(u7) = 13, f(u8) = 17, f(u9) = 19, f(u10) = 23, f(u11) = 1 . Then ef (0) = 16, ef (1) = 17.For n = 14, f(v) = 2, f(v1) = 4, f(v2) = 5, f(v3) = 6, f(v4) = 12, f(v5) = 18, f(v6) = 24, f(v7) = 8, f(v8) =10, f(v9) = 14, f(v10) = 16, f(v11) = 20, f(v12) = 22, f(v13) = 26, f(v14) = 28 and f(u1) = 7, f(u2) =3, f(u3) = 9, f(u4) = 15, f(u5) = 21, f(u6) = 27, f(u7) = 11, f(u8) = 13, f(u9) = 17, f(u10) = 19, f(u11) =23, f(u12) = 25, f(u13) = 29, f(u14) = 1. Then ef (0) = 21 = ef (1).For n = 19, f(v) = 2, f(v1) = 4, f(v2) = 5, f(v3) = 6, f(v4) = 12, f(v5) = 18, f(v6) = 24, f(v7) = 30, f(v8) =36, f(v9) = 8, f(v10) = 10, f(v11) = 14, f(v12) = 16, f(v13) = 20, f(v14) = 22, f(v15) = 26, f(v16) = 28, f(v17) =32, f(v18) = 34, f(v19) = 38 and f(u1) = 3, f(u2) = 7, f(u3) = 9, f(u4) = 15, f(u5) = 21, f(u6) = 27, f(u7) =33, f(u8) = 39, f(u9) = 11, f(u10) = 13, f(u11) = 17, f(u12) = 19, f(u13) = 23, f(u14) = 25, f(u14) =29, f(u14) = 31, f(u14) = 35, f(u14) = 37, f(u14) = 1. Then ef (0) = 29, ef (1) = 28.Now for the remaining two cases let,

s =⌊n

3

⌋, k =

⌊2n + 1

3

⌋−

⌊n3

⌋, t =

(n +

⌊2n + 1

3

⌋− 2

)−

⌈3n2

⌉,

m =⌊

2n + 12

⌋− (2 + s + t), he = largest even number not divisible by 3 ≤ 2n,

ho =largest odd number not divisible by 3 ≤ 2n + 1.f(v) = 2,f(v1) = 4, f(v2) = 5,f(v2+i) = 6i; 1 ≤ i ≤ sf(u1) = 7,f(u1+i) = 3 + 6(i− 1); 1 ≤ i ≤ k

If k = s, then f(uk+2) = 1 or f(un) = 1.Case 3: t = 0(n = 10, 12, 13, 15, 17)For m odd, consider x1 =

⌈m2

⌉, x2 = x3 = x4 =

⌊m2

⌋and for m even consider, x1 = x2 = x3 = x4 =

m

2f(vs+1+2i) = 8 + 6(i− 1); 1 ≤ i ≤ x1f(vs+2+2i) = 10 + 6(i− 1), 1 ≤ i ≤ x3f(us+1+2i) = 11 + 6(i− 1); 1 ≤ i ≤ x2f(us+2+2i) = 13 + 6(i− 1); 1 ≤ i ≤ x4

which assigns all the vertex labels for case 3.Case 4: t ≥ 1(n = 16, 18, n ≥ 20)For m odd, consider x1 = x2 = x3 =

⌊m2

⌋, x4 =

m− 32

and for m even consider, x1 =m

2, x2 = x3 = x4 =


m− 22

.

f(vs+1+2i) = 8 + 6(i− 1); 1 ≤ i ≤ x1f(vs+2+2i) = 10 + 6(i− 1), 1 ≤ i ≤ x3f(us+1+2i) = 11 + 6(i− 1); 1 ≤ i ≤ x2f(us+2+2i) = 13 + 6(i− 1); 1 ≤ i ≤ x4

For the vertices un−1, un−2, . . . , un−(t+1) we assign even numbers (not congruent 0 mod 3) in descending orderstarting from he respectively while for un,vn, vn−1, vn−2, . . . , vn−t we assign odd numbers(not congruent 0 mod3) in descending order starting from h0 respectively such that f∗(vjuj−i) or f∗(vjuj+i) do not generate edgelabel 0. Which assigns all the vertex labels for case 4.In view of the above defined labeling pattern for cases 3 and 4, we have

ef (0) =⌈

3n2

⌉and ef (1) =

⌊3n2

⌋.

Thus, we have |ef (0)− ef (1)| ≤ 1.Hence, Gn is a prime cordial graph for n ≥ 4.

Example 2.1. For the graph G20, |V (G20)| = 41 and |E(G20)| = 60. In accordance with Theorem 2.1 we haves = 6, k = 7, t = 1,m = 11, x1 = x2 = x3 = 5, x4 = 4 and using the labeling pattern described in case 4. Thecorresponding prime cordial labeling is shown in Fig. 1. It is easy to visualise that ef (0) = 30 = ef (1).

63

54

8

10

12

14

2

24

30

18

16

36

20

22

26

28

32

38

34

1

40 9

15

21

27

33

39

11

13

7

1719

23

25

29

31

35

37

41

Fig. 1

Theorem 2.2. Helm graph Hn is a prime cordial graph for every n.

Proof. Let v be the apex, v1, v2, . . . , vn be the vertices of degree 4 and u1, u2, . . . , un be the pendant verticesof Hn. Then |V (Hn)| = 2n + 1 and |E(Hn)| = 3n. To define f : V (G) → {1, 2, 3, . . . , 2n + 1}, we considerfollowing three cases.Case 1: n = 3 to 9For n = 3, f(v) = 6, f(v1) = 2, f(v2) = 4, f(v3) = 3 and f(u1) = 1, f(u2) = 7, f(u3) = 5. Then ef (0) =4, ef (1) = 5.For n = 4, f(v) = 6, f(v1) = 3, f(v2) = 2, f(v3) = 4, f(v4) = 8 and f(u1) = 1, f(u2) = 9, f(u3) = 5, f(u4) = 7.Then ef (0) = 6 = ef (1).For n = 5, f(v) = 6, f(v1) = 3, f(v2) = 2, f(v3) = 5, f(v4) = 8, f(v5) = 10 and f(u1) = 11, f(u2) = 1, f(u3) =5, f(u4) = 7, f(u5) = 9. Then ef (0) = 8, ef (1) = 7.For n = 6, f(v) = 2, f(v1) = 3, f(v2) = 6, f(v3) = 8, f(v4) = 10, f(v5) = 7, f(v6) = 11 and f(u1) = 1, f(u2) =4, f(u3) = 12, f(u4) = 5, f(u5) = 9, f(u6) = 13. Then ef (0) = 9 = ef (1).For n = 7, f(v) = 2, f(v1) = 3, f(v2) = 6, f(v3) = 4, f(v4) = 10, f(v5) = 5, f(v6) = 11, f(v7) = 1 and


f(u1) = 9, f(u2) = 8, f(u3) = 12, f(u4) = 14, f(u5) = 7, f(u6) = 13, f(u7) = 15. Then ef (0) = 11, ef (1) = 10.For n = 8, f(v) = 2, f(v1) = 6, f(v2) = 8, f(v3) = 4, f(v4) = 10, f(v5) = 5, f(v6) = 9, f(v7) = 13, f(v8) = 17and f(u1) = 3, f(u2) = 12, f(u3) = 14, f(u4) = 16, f(u5) = 7, f(u6) = 11, f(u7) = 15, f(u8) = 1. Thenef (0) = 12 = ef (1).For n = 9, f(v) = 2, f(v1) = 3, f(v2) = 6, f(v3) = 8, f(v4) = 4, f(v5) = 10, f(v6) = 5, f(v7) = 9, f(v8) =13, f(v9) = 17 and f(u1) = 1, f(u2) = 12, f(u3) = 14, f(u4) = 16, f(u5) = 18, f(u6) = 7, f(u7) = 11, f(u8) =15, f(u9) = 19. Then ef (0) = 17, ef (1) = 16.Case 2: n is even, n ≥ 10f(v) = 2, f(v1) = 10,f(v2) = 4, f(v3) = 8,f(v3+i) = 12 + 2(i− 1); 1 ≤ i ≤ n2 − 4f

(vn

2

)= 6, f

(vn

2 +1

)= 1,

f(un

2

)= 3, f

(un

2 +1

)= 2n + 1,

f(ui) = 2n− 2(i− 1); 1 ≤ i ≤ n2 − 1f(vn−i) = 5 + 4i; 0 ≤ i ≤ n2 − 1f(un−i) = 7 + 4i; 0 ≤ i ≤ n2 − 1

Case 3: n is odd, n ≥ 11f(v) = 2, f(v1) = 10,f(v2) = 4, f(v3) = 8,f(v3+i) = 12 + 2(i− 1); 1 ≤ i ≤ n−12 − 4f

(vn−1

2

)= 6, f

(vn+1

2

)= 3,

f(un+1

2

)= 1,

f(ui) = 2n− 2(i− 1); 1 ≤ i ≤ n−12f(vn−i) = 5 + 4i; 0 ≤ i < n−12f(un−i) = 7 + 4i; 0 ≤ i < n−12

In view of the above defined labeling pattern for cases 2 and 3,

If 2n− 1 ≡ 0(mod 3) then ef (0) =⌈

3n2

⌉and ef (1) =

⌊3n2

⌋,

otherwise ef (0) =⌊

3n2

⌋and ef (1) =

⌈3n2

⌉.

Thus, we have |ef (0)− ef (1)| ≤ 1.Hence, Hn is a prime cordial graph for every n.

Example 2.2. The graph H13 and its prime cordial labeling is shown in Fig. 2.

9

3

6

7

5 4

8

u1u2

v5

v6u 6

v8

v9

v10

v

10

12

14

1

v1v2

v3u3

v4

v7

u4

u 5

u 7u 8

u 9

u 10

2

27

13

11

15

17

19

23

u 11

u 12

u 13

v11

v12

v13

2624

22

20

18

16

21

25

Fig. 2

Theorem 2.3. Closed helm CHn is a prime cordial graph for n ≥ 5.


Proof. Let v be the apex, v1, v2, . . . , vn be the vertices of degree 4 and u1, u2, . . . , un be the vertices of degree 3of CHn. Then |V (CHn)| = 2n+1 and |E(CHn)| = 4n. To define f : V (G) → {1, 2, 3, . . . , 2n+1}, we considerfollowing three cases.Case 1: n = 3, 4In CH3 to satisfy the edge condition for prime cordial labeling it is essential to label six edges with label 0 andsix edges with label 1 out of twelve edges. But all the possible assignments of vertex labels will give rise to 0labels for at most four edges and 1 labels for at least eight edges. That is, |ef (0)− ef (1)| = 4 > 1. Hence, CH3is not prime cordial graph.In CH4 to satisfy the edge condition for prime cordial labeling it is essential to label eight edges with label 0and eight edges with label 1 out of sixteen edges. But all the possible assignments of vertex labels will giverise to 0 labels for at most seven edges and 1 labels for at least nine edges. That is, |ef (0) − ef (1)| = 2 > 1.Hence, CH4 is not prime cordial graph.Case 2: n = 5, 6For n = 5, f(v) = 6, f(v1) = 2, f(v2) = 4, f(v3) = 3, f(v4) = 9, f(v5) = 11 and f(u1) = 10, f(u2) = 8, f(u3) =1, f(u4) = 7, f(u5) = 5. Then ef (0) = 10 = ef (1).For n = 6, f(v) = 6, f(v1) = 1, f(v2) = 5, f(v3) = 10, f(v4) = 4, f(v5) = 12, f(v6) = 3 and f(u1) = 11, f(u2) =13, f(u3) = 8, f(u4) = 2, f(u5) = 9, f(u6) = 7. Then ef (0) = 12 = ef (1).

Case 3: n ≥ 7f(v) = 2, f(v1) = 4,f(v2) = 6, f(v3) = 3,f(v4) = 12, f(v5) = 8,f(v6) = 10,f(v6+i) = 14 + 2(i− 1); 1 ≤ i ≤ n− 6f(u1) = 1, f(u2) = 5,f(u3) = 7, f(u4) = 9,f(u5) = 13, f(u6) = 11,f(u6+i) = 15 + 2(i− 1); 1 ≤ i ≤ n− 6

In view of the above defined labeling pattern we have ef (0) = 2n = ef (1).Thus, we have |ef (0)− ef (1)| ≤ 1.Hence, CHn is a prime cordial graph for n ≥ 5.

Example 2.3. The graph CH10 and its prime cordial labeling is shown in Fig. 3.

9

3

6

7

5

4

8

1

u2

v5v6

u 6

v8

v9

v10

v

10

12

14

1

v1v2

v3

u3

v4v7u4

u 5u 7

u 8

u 9

u 10

2

16

18

20

13

11

15

17

19

21

Fig. 3

Theorem 2.4. Flower graph Fln is a prime cordial graph for n ≥ 4.


Proof. Let v be the apex, v1, v2, . . . , vn be the vertices of degree 4 and u1, u2, . . . , un be the vertices of degree2 of Fln. Then |V (Fln)| = 2n + 1 and |E(Fln)| = 4n. To define f : V (G) → {1, 2, 3, . . . , 2n + 1}, we considerfollowing four cases.

Case 1: n = 3In Fl3 to satisfy the edge condition for prime cordial labeling it is essential to label six edges with label 0 andsix edges with label 1 out of twelve edges. But all the possible assignments of vertex labels will give rise to 0labels for at most four edges and 1 labels for at least eight edges. That is, |ef (0)− ef (1)| = 4 > 1. Hence, Fl3is not prime cordial graph.Case 2: n = 4 to 9.For Fl4, f(v) = 6, f(v1) = 4, f(v2) = 2, f(v3) = 9, f(v4) = 3 and f(u1) = 7, f(u2) = 8, f(u3) = 5, f(u4) = 1.Then ef (0) = 8 = ef (1).For Fl5, f(v) = 6, f(v1) = 2, f(v2) = 4, f(v3) = 8, f(v4) = 10, f(v5) = 3 and f(u1) = 11, f(u2) = 7, f(u3) =5, f(u4) = 1, f(u5) = 9. Then ef (0) = 10 = ef (1).For Fl6, f(v) = 6, f(v1) = 2, f(v2) = 4, f(v3) = 8, f(v4) = 10, f(v5) = 12, f(v6) = 3 and f(u1) = 5, f(u2) =7, f(u3) = 9, f(u4) = 11, f(u5) = 13, f(u6) = 1. Then ef (0) = 12 = ef (1).For Fl7, f(v) = 2, f(v1) = 3, f(v2) = 12, f(v3) = 10, f(v4) = 8, f(v5) = 14, f(v6) = 4, f(v7) = 6 andf(u1) = 1, f(u2) = 5, f(u3) = 11, f(u4) = 7, f(u5) = 13, f(u6) = 15, f(u7) = 9. Then ef (0) = 14 = ef (1).For Fl8, f(v) = 2, f(v1) = 3, f(v2) = 6, f(v3) = 4, f(v4) = 8, f(v5) = 10, f(v6) = 14, f(v7) = 16, f(v8) = 12and f(u1) = 1, f(u2) = 9, f(u3) = 7, f(u4) = 5, f(u5) = 11, f(u6) = 13, f(u7) = 15, f(u8) = 17. Thenef (0) = 16 = ef (1).For Fl9, f(v) = 2, f(v1) = 3, f(v2) = 12, f(v3) = 4, f(v4) = 8, f(v5) = 10, f(v6) = 14, f(v7) = 16, f(v8) =18, f(v9) = 6 and f(u1) = 17, f(u2) = 1, f(u3) = 5, f(u4) = 7, f(u5) = 11, f(u6) = 13, f(u7) = 15, f(u8) =19, f(u9) = 9. Then ef (0) = 18 = ef (1).Case 3: n is even, n ≥ 10f(v) = 2, f(v1) = 10,f(v2) = 4, f(v3) = 8,f(v3+i) = 12 + 2(i− 1); 1 ≤ i ≤ n2 − 4f

(vn

2

)= 6,

f(vn−i) = 5 + 4i; 0 ≤ i < n2 − 2f(ui) = 2n− 2(i− 1); 1 ≤ i ≤ n2 − 1f

(un

2

)= 3,

f(un−i) = 7 + 4i; 0 ≤ i < n2 − 2For 2n + 1 ≡ 0(mod 3)f

(vn

2 +i

)= 2n + 1− 4(i− 1); 1 ≤ i ≤ 2

f(un

2 +1

)= 1, f

(un

2 +2

)= 2n− 1

For 2n + 1 ≡ 1(mod 3)f

(vn

2 +1

)= 2n− 3, f

(vn

2 +2

)= 1,

f(un

2 +1

)= 2n + 1, f

(un

2 +2

)= 2n− 1

For 2n + 1 ≡ 2(mod 3)f

(vn

2 +1

)= 2n− 1, f

(vn

2 +2

)= 2n− 3,

f(un

2 +1

)= 2n + 1, f

(un

2 +2

)= 1

Case 4: n is odd, n ≥ 11f(v) = 2, f(v1) = 10,f(v2) = 4, f(v3) = 8,f(v3+i) = 12 + 2(i− 1); 1 ≤ i ≤ n−12 − 4f

(vn−1

2

)= 6, f

(vn+1

2

)= 3,

f(vn−i) = 5 + 4i; 0 ≤ i < n−12f(ui) = 2n− 2(i− 1); 1 ≤ i ≤ n−12f(un−i) = 7 + 4i; 0 ≤ i < n−52

For 2n + 1 ≡ 0(mod 3)f

(un+1

2

)= 2n + 1, f

(un+1

2 +1

)= 1,

f(un+1

2 +2

)= 2n− 3,

For 2n + 1 ≡ 1(mod 3)


f(un+1

2

)= 2n− 3, f

(un+1

2 +1

)= 2n + 1,

f(un+1

2 +2

)= 1,

For 2n + 1 ≡ 2(mod 3)f

(un+1

2

)= 1,

f(un+1

2 +i

)= 2n + 1− 4(i− 1); 1 ≤ i ≤ 2

In view of the above defined labeling pattern we have ef (0) = 2n = ef (1).Thus, we have |ef (0)− ef (1)| ≤ 1.Hence, Fln is a prime cordial graph for n ≥ 4.

Example 2.4. The graph Fl11 and its prime cordial labeling is shown in Fig. 4.

9

3

6

75 4

8

u1 u2

v5

v6u 6

v8

v9

v10

v

10

12

14

1

v1v2

v3

u3

v4

v7

u4

u 5

u 7

u 8

u 9

u 10

213

11

15 17

19

23

u 11 v11

22 20

18

16

21

Fig. 4

3 Open problems

• To investigate necessary and sufficient conditions for a graph to admit a prime cordial labeling.

• To investigate some new graph or graph families which admit prime cordial labeling.

• To obtain forbidden subgraph(s) characterisation for prime cordial labeling.

4 Conclusion

As all the graphs are not prime cordial graphs it is very interesting and challenging as well to investigateprime cordial labeling for the graph or graph families which admit prime cordial labeling. Here we havecontributed some new results by investigating prime cordial labeling for some wheel related graphs.

5 Acknowledgement

The authors are highly thankful to the anonymous referees for their kind suggestions and comments on thefirst draft of this paper.

References

[1] L. W. Beineke and S. M. Hegde, Strongly multiplicative graphs, Discuss. Math. Graph Theory, 21(2001),63-75.


[2] I. Cahit, Cordial Graphs, A weaker version of graceful and harmonious Graphs, Ars Combinatoria,23(1987), 201-207.

[3] J. A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics, 19(2012),#DS6.

[4] R. L. Graham and N. J. A. Sloane, On additive bases and harmonious graphs, SIAM J. Alg. Disc. Meth.,1(4)(1980), 382-404.

[5] J. Gross and J. Yellen, Graph Theory and its Applications, CRC Press, 1999.

[6] Z. Liang and Z. Bai, On the odd harmonious graphs with applications, J. Appl. Math. Comput., 29(2009),105-116.

[7] R. Sridevi, K. Krishnaveni and S. Navaneethakrishnan, A novel watershed image segmentation techniqueusing graceful labeling, International Journal of Mathematics and Soft Computing, 3(1)(2013), 69-78.

[8] M. Sundaram, R. Ponraj and S. Somasundram, Prime Cordial Labeling of graphs, J. Indian Acad. Math.,27(2)(2005), 373-390.

[9] S. K. Vaidya and P. L. Vihol, Prime cordial labeling for some graphs, Modern Applied Science, 4(8)(2010),119-126.

[10] S. K. Vaidya and P. L. Vihol, Prime cordial labeling for some cycle related graphs, Int. J. of Open Problemsin Computer Science and Mathematics, 3(5)(2010), 223-232.

[11] S. K. Vaidya and N. H. Shah, Some New Families of Prime Cordial Graphs, J. of Mathematics Research,3(4)(2011), 21-30.

[12] S. K. Vaidya and N. H. Shah, Prime Cordial Labeling of Some Graphs, Open Journal of Discrete Mathe-matics, 2(2012), 11-16.

[13] V. Yegnanaryanan and P. Vaidhyanathan, Some Interesting Applications of Graph Labellings, J. Math.Comput. Sci., 2(5) (2012), 1522-1531.

Received: July 12, 2013; Accepted: July 24, 2013

UNIVERSITY PRESS

Annals of Pure and Applied Mathematics Vol. 4, No.2, 2013, 150-159 ISSN: 2279-087X (P), 2279-0888(online) Published on 27 October 2013 www.researchmathsci.org

150

Annals of

Further Results on Divisor Cordial Labeling

S. K. Vaidya1 and N. H. Shah2

1Department of Mathematics, Saurashtra University, Rajkot – 360005, Gujarat, India. E-mail: [email protected]

2Government Polytechnic, Rajkot-360003, Gujarat, India

E-mail: [email protected]

Received 23 October 2013; accepted 25 October 2013 Abstract. A divisor cordial labeling of a graph G with vertex set V is a bijection f from V to {1, 2,... | |}V such that an edge uv is assigned the label 1 if ( ) | ( )f u f v or

( ) | ( )f v f u and the label 0 otherwise, then number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph with a divisor cordial labeling is called a divisor cordial graph. In this paper we prove that helm Hn , flower graph nFl and Gear graph nG are divisor cordial graphs. Moreover we show that switching of a vertex in cycle nC , switching of a rim vertex in wheel nW and switching of the apex vertex in helm Hn admit divisor cordial labeling. Keywords: Labeling, Divisor cordial labeling, Switching of a vertex AMS Mathematics Subject Classification (2010): 05C78 1. Introduction We begin with simple, finite, connected and undirected graph ( ( ), ( ))G V G E G= with p vertices and q edges. For standard terminology and notations related to graph theory

we refer to Gross and Yellen [1] while for number theory we refer to Burton [2] . We will provide brief summary of definitions and other information which are necessary for the present investigations. Definition 1.1. If the vertices are assigned values subject to certain condition(s) then it is known as graph labeling. According to Beineke and Hegde [3] graph labeling serves as a frontier between number theory and structure of graphs. Graph labelings have enormous applications within mathematics as well as to several areas of computer science and communication networks. Yegnanaryanan and Vaidhyanathan [4] have discussed applications of edge balanced graph labeling, edge magic labeling and (1,1) edge magic graphs. For a dynamic


151

survey on various graph labeling problems along with an extensive bibliography we refer to Gallian [5]. Definition 1.2. A mapping : ( ) {0,1}f V G → is called binary vertex labeling of G and

( )f v is called the label of the vertex v of G under f . Notation 1.3. If for an edge e uv= , the induced edge labeling * : ( ) {0,1}f E G → is given by * ( ) | ( ) ( ) |f e f u f v= − . Then

*

( ) number of vertices of having label under

( ) number of vertices of having label under f

f

v i G i f

e i G i f

=

=

Definition 1.4. A binary vertex labeling f of a graph G is called a cordial labeling if | (0) (1) | 1f fv v− ≤ and | (0) (1) | 1f fe e− ≤ . A graph G is cordial if it admits cordial labeling.

The above concept was introduced by Cahit [6]. After this many labeling schemes are also introduced with minor variations in cordial theme. The product cordial labeling, total product cordial labeling and prime cordial labeling are among mention a few. The present work is focused on divisor cordial labeling. Definition 1.5. A prime cordial labeling of a graph G with vertex set ( )V G is a bijection

: ( ) {1, 2,3, , | ( ) |}f V G V G→ … and the induced function * : ( )f E G → {0,1} is defined by

* 1 , if gcd( ( ), ( )) 1;0 , otherwise.

( )f u f v

f e uv=⎧

= = ⎨⎩

which satisfies the condition | (0) (1) | 1f fe e− ≤ . A graph which admits prime cordial labeling is called a prime cordial graph.

The concept of prime cordial labeling was introduced by Sundaram et al. [7] and in the same paper they have investigated several results on prime cordial labeling. Vaidya and Vihol [8, 9] as well as Vaidya and Shah [10, 11, 12] have proved many results on prime cordial labeling.

Motivated through the concept of prime cordial labeling Varatharajan et al. [13] introduced a new concept called divisor cordial labeling which is a combination of divisibility of numbers and cordial labelings of graphs. Definition 1.6. Let ( ( ), ( ))G V G E G= be a simple graph and : {1,2,... | ( ) |}f V G→ be a bijection. For each edge uv , assign the label 1 if ( ) | ( )f u f v or ( ) | ( )f v f u and the label 0 otherwise. The function f is called a divisor cordial labeling if | (0) (1) | 1f fe e− ≤ . A graph with a divisor cordial labeling is called a divisor cordial graph.


152

In [13] authors have proved that path, cycle, wheel, star, 2,nK and 3,nK are divisor cordial graphs while nK is not divisor cordial for 7n ≥ . The divisor cordial labeling of full binary tree as well as some star related graphs are reported by Varatharajan et al. [14] while some star and bistar related graphs are proved to be divisor cordial graphs by Vaidya and Shah [15].

It is important to note that prime cordial labeling and divisor cordial labeling are two independent concepts. A graph may possess one or both of these properties or neither as exhibited below.

i) Pn (n≥ 6) is both prime cordial as proved in [7] and divisor cordial as proved in [13].

ii) C3 is not prime cordial as proved in [7] but it is divisor cordial as proved in [13]. iii) We found that a 7-regular graph with 12 vertices admits prime cordial labeling

but does not admit divisor cordial labeling. iv) Complete graph K7 is not a prime cordial as stated in Gallian [5] and not divisor

cordial as proved in [13]. Definition 1.7. The helm nH is the graph obtained from a wheel nW 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. Definition 1.8. The flower nFl is the graph obtained from a helm nH by joining each pendant vertex to the apex of the helm. It contains three types of vertices: an apex of degree 2n , n vertices of degree 4 and n vertices of degree 2. Definition 1.9. Let e=uv be an edge of the graph G and w is not a vertex of G. The edge e is called subdivided when it is replaced by edges e uw′ = and e wv′′ = . Definition 1.10. The gear graph nG is obtained from the wheel by subdividing each of its rim edge. Definition 1.11. A vertex switching vG of a graph G is the graph obtained by taking a vertex v of G , removing all the edges incident to v and adding edges joining v to every other vertex which are not adjacent to v in G . 2. Divisor Cordial Labeling of Some Wheel Related Graphs Theorem 2.1. nH is a divisor cordial graph for every n . Proof : Let v be the apex, 1 2, , , nv v v… be the vertices of degree 4 and 1 2, , , nu u u… be the pendant vertices of nH . Then | ( ) | 2 1nV H n= + and | ( ) | 3nE H n= . We define vertex labeling as : ( ) {1,2,3, ,2 1}f V G n→ … + as follows.

( ) 1,f v =

For12ni k⎡ ⎤⎢ ⎥⎢ ⎥

≤ ≤ = ,


153

Assign the labels iv and iu such that 2 ( ) ( )i if v f u= and 1( ) ( )i if v f v + . Now for remaining vertices, 1 2, , , nk kv v v+ + … and 1 2, , , nk ku u u+ + … assign the labels such that

1( ) ( )j jf v f v + where 11, ( ) ( )nk j n f v f v≤ ≤ − and ( ) ( )j jf v f u where k j n< ≤ .

In view of above labeling pattern we have, 3 3(1) , (0)2 2f fn ne e⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦

= = .

Thus, | (0) (1) | 1f fe e− ≤ . Hence, nH is a divisor cordial graph for each n . Example 2.2. The graph 13H and its divisor cordial labeling is shown in Figure 1.

Figure 1: Divisor cordial labeling of 13H Theorem 2.3. nFl is a divisor cordial graph for each n . Proof : Let v be the apex 1 2, , , nv v v… be the vertices of degree 4 and 1 2, , , nu u u… be the vertices of degree 2 of nFl . Then | ( ) | 2 1nV Fl n= + and | ( ) | 4nE Fl n= . We define vertex labeling : ( ) {1,2,3, ,2 1}f V G n→ … + as follows.

( ) 1,f v = 1( ) 2,f v = 1( ) 3,f u =

1

1

( ) 5 2( 1); 1 1( ) 4 2( 1); 1 1

i

i

f v i i nf u i i n

+

+

= + − ≤ ≤ −= + − ≤ ≤ −

In view of the above labeling pattern we have, (0) 2 (1)f fe n e= = . Thus, | (0) (1) | 1f fe e− ≤ . Hence, nFl is a divisor cordial graph for each n .


154

Example 2.4. Divisor cordial labeling of the graph 11Fl is shown in Figure 2.

Figure 2: Divisor cordial labeli

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