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ON THEOSTROWSKI-LIKETYPE INEQUALITIESFOR CONVEXMAPP INGS OF 2-PARAMETERS ONTHE CO-ORDINATES

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Far East Journal of Mathematical Sciences (FJMS) Volume …, Number …, 2012, Pages … Available online at http://pphmj.com/journals/fjms.htm Published by Pushpa Publishing House, Allahabad, INDIA

HousePublishingPushpa2012 2010 Mathematics Subject Classification: 05C78.

Keywords and phrases: ( )da, -EAMT labeling, ( )da, -SEAMT labeling, dual labeling.

Received April 13, 2012

ON SUPER EDGE ANTI MAGIC TOTAL LABELING FOR t-JOINT COPIES OF WHEEL

I W. Sudarsana1, A. Hendra1, Adiwijaya2 and D. Y. Setyawan3 1Combinatorial and Applied Mathematics Research Group Tadulako University Jalan Sukarno-Hatta Km. 9 Palu 94118, Indonesia e-mail: sudarsanaiwayan@yahoo.co.id beyourselft@yahoo.co.id

2Algorithm and Computation Research Group Institut Teknologi Telkom Jalan Telekomunikasi No. 1 Terusan Buah Batu Bandung 40257, Indonesia e-mail: adiwijaya007@yahoo.co.id

3Institut Bisnis dan Informatika Darmajaya Jalan Z. A. Pagar Alam No. 93 Bandar Lampung 35141 Indonesia e-mail: dodi.yudo@darmajaya.ac.id

Abstract

A ( )qp, -graph G is called ( )da, -edge anti magic total, ( )da, -

EAMT if there exist integers ,0>a 0≥d and a bijection

{ }qpEV +→λ ...,,2,1: ∪ such that ( ){ }ExyxywW ∈= :

( ){ },1...,,, dqadaa −++= where ( ) ( ) ( ) ( )xyyxxyw λ+λ+λ=

I W. Sudarsana, A. Hendra, Adiwijaya and D. Y. Setyawan 2

is the edge weight of xy. An ( )da, -EAMT labeling λ of G is called

super, ( )da, -SEAMT if ( ) { }....,,2,1 pV =λ In this paper, we show

that t joint copies of wheel has a ( )( )1,222 ++ tn -SEAMT labeling

for even ,4≥n and .2≥t

1. Introduction

We consider finite undirected graphs without loops and multiple edges. The notation ( )GV and ( )GE stand for the vertex set and edge set of graph

G, respectively. Let { }vue ,= ( ,shortin )uve = denote an edge connecting

vertices u and v in G. A graph nP denotes a path on n vertices. Other

standard terminologies and notations for graph theoretic ideas we follow the book of [5].

We denote by ( )qp, -graph G a graph with p vertices and q edges. A

( )qp, -graph G is called ( )da, -edge anti magic total, ( )da, -EAMT if there

exist integers 0,0 ≥> da and a bijection { }qpEV +→λ ...,,2,1: ∪ such that the set of edge-weights is

( ){ } ( ){ },1...,,,: dqadaaExyxywW −++=∈=

where ( ) ( ) ( ) ( ).xyyxxyw λ+λ+λ= We shall follow [7] to call ( ) =xyw

( ) ( ) ( )xyyx λ+λ+λ the edge-weight of xy, and W the set of edge-weights of

the graph G. In particular, an ( )da, -EAMT labeling λ of a ( )qp, -graph G

is super if ( ) { }....,,2,1 pV =λ For the rest of paper, we will denote super

( )da, -EAMT of G by ( )da, -SEAMT.

For any ( )da, -SEAMT labeling on a ( )qp, -graph G, the maximum

edge-weight is no more than ( ) ( ).1 qppp ++−+ Thus, ( ) ≤−+ dqa 1 .13 −+ qp Similarly, the minimum possible edge-weight is at least

.121 +++ p This implies that .4+≥ pa Therefore, we have

.152

−−+≤ q

qpd (1)

On Super Edge Anti Magic Total Labeling for t-joint Copies … 3

In general, for any ( )da, -EAMT labeling on a ( )qp, -graph G, the

maximum edge-weight is no more than ( ) ( ) ( ).12 qpqpqp ++−++−+

Thus, ( ) .3331 −+≤−+ qpdqa Similarly, the minimum possible edge-

weight is at least .321 ++ Consequently, .6≥a So, we have

.1933

−−+≤ q

qpd (2)

A number of classification studies on ( )da, -SEAMT (resp. ( )da, -

EAMT) for connected graphs has been extensively investigated. For instances, in [2], Bača et al. showed that wheel nW has a ( )da, -SEAMT

labeling if and only if 1=d and ( );4mod1≡n Fan nF has an ( )da, -

SEAMT if 62 ≤≤ n and { }.2,1,0∈d Ngurah and Baskoro [6] proved that

for every Petersen graph ( ) ,21,3,, nmnmnP ≤≤≥ has a ( )1,24 +n -

SEAMT labeling. More results concerning SEAMT graphs can be seen in [1] and survey paper by Gallian [4]. People also consider how to construct a new (bigger) ( )da, -SEAMT graphs from some known (smaller) ( )da, -SEAMT

graphs. These constructions are proposed by inserting some new pendant edges and points, see for instance [3, 7, 8, 10, 11]. However, the ( )da, -

SEAMT labeling for t-joint copies of wheel, ( ),, ntW is still open.

In this paper, we show that ( )ntW , has a ( )( )1,222 ++ tn -SEAMT

labeling for even ,4≥n and .2≥t

2. Preliminary Theorems

The properties of ( )da, -SEAMT labeling of graph proposed in the

following theorems will be useful in the next section. Given any ( )da, -

EAMT labeling λ on a ( )qp, -graph G. Then, its dual labeling λ′ can be

defined [12] by

I W. Sudarsana, A. Hendra, Adiwijaya and D. Y. Setyawan 4

( ) ( )xqpx λ−++=λ′ 1 for any vertex x, and

( ) ( )xyqpxy λ−++=λ′ 1 for any edge xy.

By using this duality, Wallis et al. [12] established the following theorem.

Theorem A. (Wallis et al. [12]). If a ( )qp, -graph G has an ( )da, -

EAMT labeling, then G has an ( )da ,′ -EAMT labeling as its dual with

( ) .1333 dqaqpa −−−++=′

Theorem B. (Sudarsana et al. [9]). Let λ be a ( )da, -SEAMT labeling of

a ( )qp, -graph G. Then the labeling λ′ defined:

( ) ( ) Vxxpx ∈∀λ−+=λ′ ,1 1 and

( ) ( ) Exyxyqpxy ∈∀λ−++=λ′ ,12 1

is a ( )da ,′ -SEAMT labeling of G with ( ) .134 dqaqpa −−−++=′

The labeling λ′ is called a dual ( )da, -SEAMT labeling of λ on G.

Moreover, the dual labeling is called selfdual, if .aa ′=

3. The Main Result

Let nW be a wheel on 1+n vertices with 0v as the hub and cycle

1321 vvvvv n as the rim. The t-joint copies of wheel, denoted by ( ),,ntW is a

graph obtained by taking t copies of nW and joining every hub of the wheels

by an edge such that forming a path .tP We provide an example of ( )6,4W

in Figure 1:

Figure 1. The 4-joint copies of ( )., 6,46 WW

On Super Edge Anti Magic Total Labeling for t-joint Copies … 5

In general, we denote that ( ) { }nitjvWV jint ≤≤≤≤= 0,1:,, and

( ) { },,, ,,,,oj

ojijnjint eeeeWE = where

≤≤=

≤≤−≤≤=

+

,1,,

;1,11,

,1,

,1,, tjnivv

tjnivve

jjn

jijiji

,1,1,,,0, nitjvve jijo

ji ≤≤≤≤=

.11,1,0,0 −≤≤= + tjvve jjoj

By (1) and (2), we have: for every even ,4≥n and 2≥t is an integer, there is no ( )da, -SEAMT labeling of ( )ntW , with ;3≥d and there is no

( )da, -EAMT labeling of ( )ntW , with .5≥d

The following theorem deals with ( )( )1,222 ++ tn -SEAMT labeling of

( )., ntW

Theorem 1. Let 4≥n and 2≥t be integers, and n is even. Then the graph ( )ntW , has a ( )( )1,222 ++ tn -SEAMT labeling. This type of labeling

is selfdual.

Proof. Label the vertices and edges of ( )ntW , in the following way:

( )

( )

( )

≤≤≤≤++

≤≤=+

≤≤≤≤+−

=λ

,1,22,

;1,0,2

;1,21,

,

tjninjit

tjijnt

tjnijtji

v ji

( )

( )

( ) ( )

≤≤=−+

+

≤≤−≤≤+−+−+

≤≤≤≤−+−+

=λ

,1,,1223

;1,122,112

;1,21,122

,

tjnijtn

tjninjtin

tjnijtin

e ji

I W. Sudarsana, A. Hendra, Adiwijaya and D. Y. Setyawan 6

( )( )

( ) ( )

≤≤≤≤+−+−+

≤≤≤≤−+−+=λ

,1,22,1224

;1,21,1233,

tjninjtin

tjnijtineo

ji

( ) ( ) .1,23 tjjtneoj ≤≤−+=λ

Now, we construct the set of edge-weights W in the following way:

{ ( ) ( ) ( )},,, ,,oj

ojiji ewewewW =

where

( )( ) ( ) ( )

( ) ( ) ( )

≤≤=λ+λ+λ

≤≤−≤≤λ+λ+λ=

+

,1,,

;1,11,

,1,,

,1,,,

tjnivev

tjnivevew

jjnjn

jijijiji

( ) ( ) ( ) ( ) ;1,1,,,,0, tjnivevew jio

jijo

ji ≤≤≤≤λ+λ+λ=

( ) ( ) ( ) ( ) .11,1,0,0 −≤≤λ+λ+λ= + tjvevew jojj

oj

Therefore, we obtain that the edge-weights are

( )

( )

( ) ( )

≤≤=++

+

≤≤−≤≤+++++

≤≤=++

+

≤≤−≤≤++++

=

,1,,1225

;1,122,122

;1,2,1245

;1,121,112

,

tjnijtn

tjninjtin

tjnijtn

tjnijtin

ew ji

( )( )

≤≤≤≤

+++

−+

≤≤≤≤++

−+

=,1,2

2,2249

;1,21,1257

,tjninjtin

tjnijtin

ew oji

( ) ( ) .11,124 −≤≤+++= tjjtnew oj

On Super Edge Anti Magic Total Labeling for t-joint Copies … 7

Finally, we have that the set of edge-weights is

( ) ( ) ( ) ( ){ }.34,134...,,322,222 tntntntnW +−+++++=

We can see that the set W consists of the consecutive integers starting from ( ) .222 ++= tna By Theorem B, it can be verified that this labeling

is selfdual. The proof is now complete.

Figure 2. A (44, 1)-SEAMT labeling of ( )6,3W obtained from Theorem 1.

By Theorem A, we have the following corollary.

Corollary 1. For every even 4≥n and 2≥t is an integer, the graph

( )ntW , has an ( )( )1,35 tn + -EAMT labeling.

To conclude this paper, let us present two open problems to work on. Construct, if there exists

Figure 3. An (99, 1)-EAMT labeling of ( )6,3W obtained from Theorems 1

and A.

1. An ( )da, -SEAMT labeling of ( )ntW , with { },1,0∈d for even

,4≥n and .2≥t

I W. Sudarsana, A. Hendra, Adiwijaya and D. Y. Setyawan 8

2. An ( )da, -SEAMT labeling of ( )ntW , with { },2,1,0∈d for odd

,5≥n and .2≥t

Acknowledgement

The first and second authors gratefully acknowledge to the Directorate General of Higher Education of Indonesia (DIKTI), State Ministry of Education and Culture for financial support under Penelitian Fundamental: 125.C/un 28.2/PL/2012.

References

[1] M. Bača, Y. Lin, M. Miller and R. Simanjuntak, New constructions of magic and antimagic graph labelings, Utilitas Math. 60 (2001), 229-239.

[2] M. Bača, Y. Lin, M. Miller and M. Z. Youssef, Edge-antimagic graphs, Discrete Math. 307 (2007), 1232-1244.

[3] E. T. Baskoro, I W. Sudarsana and Y. M. Cholily, How to construct new super edge-magic graphs from some old ones, J. Indones. Math. Soc. 11(2) (2005), 156-162.

[4] J. A. Gallian, A dynamic survey of graph labellings, Electron. J. Combin. 18 (2011), #DS6.

[5] N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, San Diego, 1994.

[6] A. A. G. Ngurah and E. T. Baskoro, On magic and antimagic total labeling of generalized Petersen graph, Utilitas Math. 63 (2003), 97-107.

[7] Kiki A. Sugeng and Mirka Miller, Relationship between adjacency matrices and super ( )da, -edge-antimagic-total labeling of graphs, J. Combin. Math. Combin.

Comput. 55 (2005), 71-82.

[8] I W. Sudarsana, E. T. Baskoro, D. Izmaimusa and H. Assiyatun, Creating new super edge-magic total labelings from old ones, J. Combin. Math. Combin. Comput. 55 (2005), 83-90.

[9] I W. Sudarsana, E. T. Baskoro, D. Izmaimusa and H. Assiyatun, On super ( )da, -

edge antimagic total labeling of disconnected graphs, J. Combin. Math. Combin. Comput. 55 (2005), 149-158.

On Super Edge Anti Magic Total Labeling for t-joint Copies … 9

[10] I W. Sudarsana, E. T. Baskoro, S. Uttunggadewa and D. Izmaimusa, An expansion technique on super edge-magic total graphs, Ars Combin. 91 (2009), 231-241.

[11] I W. Sudarsana, E. T. Baskoro, S. Uttunggadewa and D. Izmaimusa, Expansion techniques on the super edge anti magic total graphs, J. Combin. Math. Combin. Comput. 71 (2009), 189-199.

[12] W. D. Wallis, E. T. Baskoro, M. Miller and Slamin, Edge-magic total labellings, Australasian J. Combin. 22 (2000), 177-190.