downloads.hindawi.comdownloads.hindawi.com/journals/jam/2013/896815.pdf · On Super (,) -Edge-Antimagic Total Labeling of Special Types of Crown Graphs ... form an arithmetic progression

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 896815 6 pageshttpdxdoiorg1011552013896815

Research ArticleOn Super (119886 119889)-Edge-Antimagic Total Labeling of Special Typesof Crown Graphs

Himayat Ullah1 Gohar Ali1 Murtaza Ali1 and Andrea SemaniIovaacute-Fe^ovIiacutekovaacute2

1 FAST-National University of Computer and Emerging Sciences Peshawar 25000 Pakistan2Department of Applied Mathematics and Informatics Technical University 042 00 Kosice Slovakia

Correspondence should be addressed to Gohar Ali goharalinuedupk

Received 8 April 2013 Accepted 9 June 2013

Academic Editor Zhihong Guan

Copyright copy 2013 Himayat Ullah et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

For a graph 119866 = (119881 119864) a bijection 119891 from 119881(119866) cup 119864(119866) rarr 1 2 |119881(119866)| + |119864(119866)| is called (119886 119889)-edge-antimagic total ((119886 119889)-EAT) labeling of 119866 if the edge-weights 119908(119909119910) = 119891(119909) + 119891(119910) + 119891(119909119910) 119909119910 isin 119864(119866) form an arithmetic progression starting from 119886and having a common difference 119889 where 119886 gt 0 and 119889 ge 0 are two fixed integers An (119886 119889)-EAT labeling is called super (119886 119889)-EATlabeling if the vertices are labeled with the smallest possible numbers that is119891(119881) = 1 2 |119881(119866)| In this paper we study super(119886 119889)-EAT labeling of cycles with some pendant edges attached to different vertices of the cycle

1 Introduction

All graphs considered here are finite undirected and withoutloops andmultiple edges Let119866 be a graph with the vertex set119881 = 119881(119866) and the edge set 119864 = 119864(119866) For a general referenceof the graph theoretic notions see [1 2]

A labeling (or valuation) of a graph is a map thatcarries graph elements to numbers usually to positive ornonnegative integers In this paper the domain of the mapis the set of all vertices and all edges of a graph Such type oflabeling is called total labeling Some labelings use the vertex-set only or the edge-set only and we will call them vertexlabelings or edge labelings respectively The most completerecent survey of graph labelings can be seen in [3 4]

A bijection 119892 119881(119866) rarr 1 2 |119881(119866)| is called (119886 119889)-edge-antimagic vertex ((119886 119889)-EAV) labeling of 119866 if the set ofedge-weights of all edges in 119866 is equal to the set 119886 119886 + 119889 119886 +2119889 119886 + (|119864(119866)| minus 1)119889 where 119886 gt 0 and 119889 ge 0 are twofixed integers The edge-weight 119908119892(119909119910) of an edge 119909119910 isin 119864(119866)under the vertex labeling 119892 is defined as the sum of the labelsof its end vertices that is119908119892(119909119910) = 119892(119909)+119892(119910) A graph thatadmits (119886 119889)-EAV labeling is called an (119886 119889)-EAV graph

A bijection119891 119881(119866)cup119864(119866) rarr 1 2 |119881(119866)|+ |119864(119866)|

is called (119886 119889)-edge-antimagic total ((119886 119889)-EAT) labeling of119866

if the set of edge-weights of all edges in119866 forms an arithmeticprogression starting from 119886with the difference 119889 where 119886 gt 0and 119889 ge 0 are two fixed integers The edge-weight 119908119891(119909119910) ofan edge 119909119910 isin 119864(119866) under the total labeling 119891 is defined asthe sum of the edge label and the labels of its end verticesThis means that 119908119891(119909119910) = 119891(119909) + 119891(119910) + 119891(119909119910) 119909119910 isin119864(119866) = 119886 119886+119889 119886+2119889 119886+(|119864(119866)|minus1)119889Moreover if thevertices are labeled with the smallest possible numbers thatis 119891(119881) = 1 2 |119881(119866)| then the labeling is called super(119886 119889)-edge-antimagic total (super (119886 119889)-EAT) A graph thatadmits an (119886 119889)-EAT labeling or a super (119886 119889)-EAT labelingis called an (119886 119889)-EAT graph or a super (119886 119889)-EAT graphrespectively

The super (119886 0)-EAT labelings are usually called superedge-magic see [5ndash8] Definition of super (119886 119889)-EAT labelingwas introduced by Simanjuntak et al [9] This labeling is anatural extension of the notions of edge-magic labeling see[7 8] Many other researchers investigated different types ofantimagic graphs For example see Bodendiek and Walther[10] Hartsfield and Ringel [11]

In [9] Simanjuntak et al defined the concept of (119886 119889)-EAV graphs and studied the properties of (119886 119889)-EAV labelingand (119886 119889)-EAT labeling and gave constructions of (119886 119889)-EATlabelings for cycles and paths Baca et al [12] presented some

2 Journal of Applied Mathematics

relation between (119886 119889)-EAT labeling and other labelingsnamely edge-magic vertex labeling and edge-magic totallabeling

In this paper we study super (119886 119889)-EAT labeling of theclass of graphs that can be obtained from a cycle by attachingsome pendant edges to different vertices of the cycle

2 Basic Properties

Let us first recall the known upper bound for the parameter119889 of super (119886 119889)-EAT labeling Assume that a graph 119866 hasa super (119886 119889)-EAT labeling 119891 119891 119881(119866) cup 119864(119866) rarr

1 2 |119881(119866)| + |119864(119866)| The minimum possible edge-weight in the labeling 119891 is at least 1 + 2 + (|119881(119866)| + 1) =|119881(119866)| + 4 Thus 119886 ge |119881(119866)| + 4 On the other hand themaximum possible edge-weight is at most (|119881(119866)| minus 1) +|119881(119866)| + (|119881(119866)| + |119864(119866)|) = 3|119881(119866)| + |119864(119866)| minus 1 So

119886 + (|119864 (119866)| minus 1) 119889 le 3 |119881 (119866)| + |119864 (119866)| minus 1 (1)

119889 le2 |119881 (119866)| + |119864 (119866)| minus 5

|119864 (119866)| minus 1 (2)

Thus we have the upper bound for the difference 119889 Inparticular from (2) it follows that for any connected graphwhere |119881(119866)| minus 1 le |119864(119866)| the feasible value 119889 is no morethan 3

The next proposition proved by Baca et al [12] gives amethod on how to extend an edge-antimagic vertex labelingto a super edge-antimagic total labeling

Proposition 1 (see [12]) If a graph 119866 has an (119886 119889)-EAVlabeling then

(i) 119866 has a super (119886+ |119881(119866)| +1 119889+1)-EAT labeling and

(ii) 119866 has a super (119886+ |119881(119866)|+ |119864(119866)| 119889minus1)-EAT labeling

Thefollowing lemmawill be useful to obtain a super edge-antimagic total labeling

Lemma 2 (see [13]) Let A be a sequence A = 119888 119888 + 1 119888 +

2 119888 + 119896 119896 even Then there exists a permutation Π(A) ofthe elements ofA such thatA +Π(A) = 2119888 + 1198962 2119888 + 1198962 +1 2119888 + 1198962 + 2 2119888 + 31198962 minus 1 2119888 + 31198962

Using this lemma we obtain that if 119866 is an (119886 1)-EAVgraph with odd number of edges then 119866 is also super (1198861015840 1)-EAT

3 Crowns

If 119866 has order 119899 the corona of 119866 with119867 denoted by 119866 ⊙ 119867is the graph obtained by taking one copy of 119866 and 119899 copies of119867 and joining the 119894th vertex of119866with an edge to every vertexin the 119894th copy of 119867 A cycle of order 119899 with an 119898 pendantedges attached at each vertex that is 119862119899 ⊙ 1198981198701 is called an119898-crown with cycle of order 119899 A 1-crown or only crown isa cycle with exactly one pendant edge attached at each vertex

of the cycle For the sake of brevity we will refer to the crownwith cycle of order 119899 simply as the crown if its cycle order isclear from the context Note that a crown is also known in theliterature as a sun graph In this section we will deal with thegraphs related to 1-crown with cycle of order 119899

Silaban and Sugeng [14] showed that if the 119898-crown119862119899 ⊙1198981198701 is (119886 119889)-EAT then 119889 le 5 They also describe (119886 119889)-EAT labeling of the 119898-crown for 119889 = 2 and 119889 = 4 Note thatthe (119886 2)-EAT labeling of119898-crown presented in [14] is super(119886 2)-EAT Moreover they proved that if 119898 equiv 1(mod 4) 119899and 119889 are odd there is no (119886 119889)-EAT labeling for 119862119899 ⊙ 1198981198701They also proposed the following open problem

Open Problem 1 (see [14]) Find if there is an (119886 119889)-EATlabeling 119889 isin 1 3 5 for119898-crown graphs 119862119899 ⊙ 1198981198701

Figueroa-Centeno et al [15] proved that the 119898-crowngraph has a super (119886 0)-EAT labeling

Proposition 3 (see [15]) For every two integers 119899 ge 3 and119898 ge 1 the119898-crown 119862119899 ⊙ 1198981198701 is super (119886 0)-EAT

According to inequality (2) we have that if the crown119862119899 ⊙ 1198701 is super (119886 119889)-EAT then 119889 le 2 Immediately fromProposition 3 and the results proved in [14] we have that thecrown 119862119899 ⊙ 1198701 is super (119886 0)-EAT and super (119886 2)-EAT forevery positive integer 119899 ge 3 Moreover for 119899 odd the crownis not (119886 1)-EAT In the following theorem we prove that thecrown is super (119886 1)-EAT for 119899 even Thus we partially givean answer to Open Problem 1

Theorem 4 For every even positive integer 119899 119899 ge 4 the crown119862119899 ⊙ 1198701 is super (119886 1)-EAT

Proof Let 119881(119862119899 ⊙ 1198701) = V1 V2 V119899 1199061 1199062 119906119899be the vertex set and 119864(119862119899 ⊙ 1198701) =

V1V2 V2V3 V119899V1 V11199061 V21199062 V119899119906119899 be the edge setof 119862119899 ⊙ 1198701 see Figure 1

We define a labeling 119891 119891 119881(119862119899 ⊙ 1198701) cup 119864(119862119899 ⊙ 1198701) rarr1 2 4119899 as follows

119891 (V119894) =

119894 + 119899 for 119894 = 1 2 1198992

119894 for 119894 = 1198992+ 1

119899

2+ 2 119899

119891 (119906119894) =

119894 for 119894 = 1 2 1198992

119894 + 119899 for 119894 = 1198992+ 1

119899

2+ 2 119899

119891 (V119894V119894+1) =

7119899

2+ 1 minus 119894 for 119894 = 1 2 119899

2

9119899

2+ 1 minus 119894 for 119894 = 119899

2+ 1

119899

2+ 2 119899 minus 1

119891 (V1V119899) =7119899

2+ 1

119891 (V119894119906119894) = 3119899 + 1 minus 119894 for 119894 = 1 2 119899(3)

Journal of Applied Mathematics 3

u1

un

u2

u3

u4

unminus1

1

n

nminus1

2

3

4

⋱

Figure 1 The crown 119862119899⊙ 1198701

It is easy to check that 119891 is a bijection For the edge-weights under the labeling 119891 we have

119908119891 (V1V119899) =11119899

2+ 2

119908119891 (V119894V119894+1) =

11119899

2+ 2 + 119894 for 119894 = 1 2 119899

2minus 1

9119899

2+ 2 + 119894 for 119894 = 119899

2119899

2+ 1 119899 minus 1

119908119891 (V119894119906119894) = 4119899 + 1 + 119894 for 119894 = 1 2 119899(4)

Thus the edge-weights are distinct number from the set 4119899+2 4119899 + 3 6119899 + 1 This means that 119891 is a super (119886 1)-EATlabeling of the crown 119862119899 ⊙ 1198701

In [16] was proved the following

Proposition 5 (see [16]) Let 119866 be a super (119886 1)-EAT graphThen the disjoint union of arbitrary number of copies of119866 thatis 119904119866 119904 ge 1 also admits a super (1198861015840 1)-EAT labeling

In [17] Figueroa-Centeno et al proved the following

Proposition 6 (see [17]) If 119866 is a (super) (119886 0)-EAT bipartiteor tripartite graph and 119904 is odd then 119904119866 is (super) (1198861015840 0)-EAT

For (super) (119886 2)-EAT labeling for disjoint union ofcopies of a graph was shown the following


Using the above mentioned results we immediatelyobtain the following theorem

Theorem 8 Let 119899 be an even positive integer 119899 ge 4 Thenthe disjoint union of arbitrary number of copies of the crown119862119899 ⊙ 1198701 that is 119904(119862119899 ⊙ 1198701) 119904 ge 1 admits a super (119886 1)-EATlabeling

Since the crown 119862119899 ⊙ 1198701 is either a bipartite graph for119899 even or a tripartite graph for 119899 odd thus we have thefollowing

Theorem9 Let 119899 be a positive integer 119899 ge 3Then the disjointunion of odd number of copies of the crown 119862119899 ⊙ 1198701 that is119904(119862119899 ⊙ 1198701) 119904 ge 1 admits a super (119886 119889)-EAT labeling for 119889 isin0 2

According to these results we are able to give partiallypositive answers to the open problems listed in [19]

Open Problem 2 (see [19]) For the graph 119904(119862119899 ⊙ 1198981198701) 119904119899even and 119898 ge 1 determine if there is a super (119886 119889)-EATlabeling with 119889 isin 0 2

Open Problem 3 (see [19]) For the graph 119904(119862119899 ⊙1198981198701) 119904 oddand 119899(119898 + 1) even determine if there is a super (119886 1)-EATlabeling

4 Graphs Related to Crown Graphs

Let us consider the graph obtained from a crown graph 119862119899 ⊙1198701 by deleting one pendant edge

Theorem 10 For 119899 odd 119899 ge 3 the graph obtained from acrown graph119862119899⊙1198701 by removing a pendant edge is super (119886 119889)-EAT for 119889 isin 0 1 2

Proof Let119866 be a graph obtained from a crown graph119862119899 ⊙1198701by removing a pendant edge Without loss of generality wecan assume that the removed edge is V11199061 Other edges andvertices we denote in the same manner as that of Theorem 4Thus |119881(119866)| = 2119899 minus 1 and |119864(119866)| = 2119899 minus 1 It follows from (2)that 119889 le 2

Define a vertex labeling 119892 as follows

119892 (V119894) = 119894 for 119894 = 1 3 119899119894 + 119899 minus 1 for 119894 = 2 4 119899 minus 1

119892 (119906119894) = 119894 + 119899 minus 1 for 119894 = 3 5 119899119894 for 119894 = 2 4 119899 minus 1

(5)

It is easy to see that labeling 119892 is a bijection from the vertexset to the set 1 2 2119899minus1 For the edge-weights under thelabeling 119892 we have

119908119892 (V1V119899) = 1 + 119899

119908119892 (V119894V119894+1) = 2119894 + 119899 for 119894 = 1 2 119899 minus 1

119908119892 (V119894119906119894) = 2119894 + 119899 minus 1 for 119894 = 2 3 119899

(6)

Thus the edge-weights are consecutive numbers 119899 + 1 119899 +2 3119899 minus 1 This means that 119892 is the (119886 1)-EAV labeling of119866 According to Proposition 1 the labeling 119892 can be extendedto the super (1198860 0)-EAT and the super (1198862 2)-EAT labelingof 119866 Moreover as the size of 119866 is odd |119864(119866)| = 2119899 minus 1 andaccording to Lemma 2 we have that 119866 is also super (1198861 1)-EAT


Note that we found some (119886 1)-EAV labelings for thegraphs obtained from a crown graph 119862119899 ⊙ 1198701 by removinga pendant edge only for small size of 119899 where 119899 is evenIn Figure 2 there are depicted (119886 1)-EAV labelings of thesegraphs for 119899 = 4 6 8 However we propose the followingconjecture

Conjecture 11 The graph obtained from a crown graph 119862119899 ⊙1198701 119899 ge 3 by removing a pendant edge is super (119886 119889)-EAT for119889 isin 0 1 2

Now we will deal with the graph obtained from a crowngraph 119862119899 ⊙ 1198701 by removing two pendant edges at distance 1and at distance 2

First consider the case when we remove two pendantedges at distance 1 Let us consider the graph119866with the (119886 1)-EAV labeling119892defined in the proof ofTheorem 10 It is easy tosee that the vertex 119906119899 is labeled with themaximal vertex label119892(119906119899) = 2119899 minus 1 Also the edge-weight of the edge V119899119906119899 is themaximal possible 119908119892(V119899119906119899) = 119892(V119899) + 119892(119906119899) = (2119899 minus 1) + 119899 =3119899 minus 1 Thus it is possible to remove the edge V119899119906119899 from thegraph 119866 we denote the graph by 119866119903 and for the labeling 119892restricted to the graph 119866119903 we denote it by 119892119903 Clearly 119903119903 is(119886 1)-EAV labeling of 119866119903 According to Proposition 1 fromthe labeling 119892119903 we obtain the super (1198860 0)-EAT and the super(1198862 2)-EAT labeling of119866119903 Note that as the size of 119866119903 is even|119864(119866119903)| = 2119899 minus 2 the labeling 119892119903 can not be extended to asuper (1198861 1)-EAT labeling of 119866119903

Theorem 12 For odd 119899 119899 ge 3 the graph obtained from acrown graph119862119899⊙1198701 by removing two pendant edges at distance1 is super (119886 119889)-EAT for 119889 isin 0 2

Next we show that if we remove from a crown graph119862119899⊙1198701 119899 ge 5 two pendant edges at distance 2 the resulting graphis super (119886 119889)-EAT for 119889 isin 0 2


Proof Let 119866 be a graph obtained from a crown graph 119862119899 ⊙1198701 by removing two pendant edges at distance 2 Let 119881(119866) =V1 V2 V119899 1199062 1199063 119906119899minus2 119906119899 be the vertex set and 119864(119866) =V1V2 V2V3 V119899V1 V21199062 V31199063 V119899minus2119906119899minus2 V119899119906119899 be the edgeset of 119866 Thus |119881(119866)| = 2119899 minus 2 and |119864(119866)| = 2119899 minus 2 Using (2)gives 119889 le 2

Define a vertex labeling 119892 in the following way

119892 (V119894) =

119894 + 1

2 for 119894 = 1 3 119899

119899 + 119894 + 1

2 for 119894 = 2 4 119899 minus 1

119892 (119906119894) =

3119899 minus 4 + 119894

2 for 119894 = 3 5 119899

2119899 + 119894

2 for 119894 = 2 4 119899 minus 3

(7)

1

3

2

75

6

410

8

9

4

57

6

2

5

3

8

1

4

9

10

7

11

614

13

15

12

10 7

8

119

5

6

3

7

4

11

1

5

2

6

10

14

9

15

8

12

1317

18

20

16

13

14

1910

11

15

126

7

8

9

Figure 2 The (119886 1)-EAV labelings for the graphs obtained from acrown graph 119862

119899⊙ 1198701by removing one pendant edge for 119899 = 4 6 8

The edge-weights under the corresponding labelings are depicted initalic

It is not difficult to check that the labeling119892 is a bijection fromthe vertex set of119866 to the set 1 2 2119899minus2 and that the edge-weights under the labeling 119892 are consecutive numbers (119899 +3)2 (119899+5)2 (5119899minus3)2Thus119892 is the (119886 1)-EAV labelingof 119866 According to Proposition 1 it is possible to extend thelabeling 119892 to the super (1198860 0)-EAT and the super (1198862 2)-EATlabelings of 119866

Result in the following theorem is based on the PetersenTheorem

Proposition 14 (Petersen Theorem) Let 119866 be a 2119903-regulargraph Then there exists a 2-factor in 119866


Notice that after removing edges of the 2-factor guaran-teed by the PetersenTheorem we have again an even regulargraph Thus by induction an even regular graph has a 2-factorization

The construction in the following theorem allows to finda super (119886 1)-EAT labeling of any graph that arose from aneven regular graph by adding even number of pendant edgesto different vertices of the original graph Notice that theconstruction does not require the graph to be connected

Theorem 15 Let 119866 be a graph that arose from a 2119903-regulargraph119867 119903 ge 0 by adding 119896 pendant edges to 119896 different verticesof119867 0 le 119896 le |119881(119867)| If 119896 is an even integer then the graph 119866is super (119886 1)-EAT

Proof Let119866 be a graph that arose from a 2119903-regular graph119867119903 ge 0 by adding 119896 pendant edges to 119896 different vertices of1198670 le 119896 le |119881(119867)| and let119898 = |119881(119867)|minus119896Thus |119881(119866)| = 2119896+119898|119864(119866)| = 119896 + 119903(119896 + 119898) and |119864(119867)| = 119903(119896 + 119898)

Let 119896 be an even integer Let us denote the pendant edgesof119866 by symbols 1198901 1198902 119890119896Wedenote the vertices of119866 suchthat

119890119894 = V119894V119896+119898+119894 119894 = 1 2 119896 (8)

and moreover

V1198962+1 V1198962+2 V119896 V119896+119898+1 V119896+119898+2 V31198962+119898 sub 119881 (119867)

(9)

We denote the remaining 119898 vertices of 119881(119867) arbitrarily bythe symbols V119896+1 V119896+2 V119896+119898

By the PetersenTheorem there exists a 2-factorization of119867 We denote the 2-factors by 119865119895 119895 = 1 2 119903 Withoutloss of generality we can suppose that 119881(119867) = 119881(119865119895) forall 119895 119895 = 1 2 119903 and 119864(119866) = cup119903

119895=1119864(119865119895) cup 1198901 1198902 119890119896

Each factor 119865119895 is a collection of cycles We order and orientthe cycles arbitrarily such that the arcs form oriented cyclesNow we denote by the symbol 119890out

119895(V119905) the unique outgoing

arc that forms the vertex V119905 in the factor 119865119895 119905 = 1198962 +

1 1198962 + 2 31198962 + 119898 Note that each edge is denoted bytwo symbols

We define a total labeling 119891 of 119866 in the following way

119891 (V119894) = 119894 for 119894 = 1 2 2119896 + 119898

119891 (119890119894) = 3119896 + 119898 + 1 minus 119894 for 119894 = 1 2 119896

119891 (119890out119895(V119905)) =

7119896

2+ 119898 + 1 minus 119891 (V119905) + 119895 (119896 + 119898)

for 119905 = 1198962+ 1

119896

2+ 2

3119896

2+ 119898 119895 = 1 2 119903

(10)

It is easy to see that the vertices are labeled by the first 2119896+119898 integers the edges 1198901 1198902 119890119896 by the next 119896 labels and theedges of119867 by consecutive integers starting at 3119896+119898+1Thus119891 is a bijection 119881(119866) cup 119864(119866) rarr 1 2 3119896 + 119898 + 119903(119896 + 119898)

It is not difficult to verify that 119891 is a super (4119896 + 2119898 +2 1)-EAT labeling of 119866 For the weights of the edges 119890119894 119894 =1 2 119896 we have

119908119891 (119890119894) = 119891 (V119894) + 119891 (V119896+119898+1) + 119891 (119890119894)

= 119894 + (119896 + 119898 + 119894) + (3119896 + 119898 + 1 minus 119894)

= 4119896 + 2119898 + 1 + 119894

(11)

Thus the edge-weights are the numbers

4119896 + 2119898 + 2 4119896 + 2119898 + 3 5119896 + 2119898 + 1 (12)

For convenience we denote by V119904 the unique vertex such thatV119905V119904 = 119890

out119895(V119905) in 119865119895 where 119904 isin 1198962+1 1198962+2 31198962+119898

The weights of the edges in 119865119895 119895 = 1 2 119903 are

119908119891 (119890out119895(V119905)) = 119891 (V119905) + 119891 (V119904) + 119891 (119890

out119895(V119905))

= 119891 (V119905) + 119891 (V119904)

+ (7119896

2+ 119898 + 1 minus 119891 (V119905) + 119895 (119896 + 119898))

=7119896

2+ 119898 + 1 + 119895 (119896 + 119898) + 119891 (V119904)

(13)

for all 119905 = (1198962)+1 (1198962)+2 (31198962)+119898 and 119895 = 1 2 119903Since 119865119895 is a factor in 119867 it holds 119891(V119904) V119904 isin 119865119895 = 1198962 +1 1198962 + 2 31198962 + 119898 Hence we have that the set of theedge-weights in the factor 119865119895 is

4119896 + 119898 + 2 + 119895 (119896 + 119898) 4119896 + 119898 + 3 + 119895 (119896 + 119898)

5119896 + 2119898 + 1 + 119895 (119896 + 119898)

(14)

and thus the set of all edge-weights in 119866 under the labeling119891 is

4119896 + 2119898 + 2 4119896 + 2119898 + 3 5119896 + 2119898 + 1 + 119903 (119896 + 119898)

(15)

We conclude the paper with the result that immediatelyfollows from the previous theorem

Corollary 16 Let 119866 be a graph obtained from a crown graph119862119899 ⊙ 1198701 119899 ge 3 by deleting any 119896 pendant edges 0 le 119896 le 119899 If119899 minus 119896 is even then 119866 is a super (119886 1)-EAT graph

Acknowledgments

This research is partially supported by FAST-National Uni-versity of Computer and Emerging Sciences Peshawar andHigher Education Commission of Pakistan The researchfor this paper was also supported by Slovak VEGA Grant1013012


References

[1] W D Wallis Magic Graphs Birkhauser Boston Mass USA2001

[2] D BWestAn Introduction toGraphTheory PrenticeHall 1996[3] M Baca andMMiller Super Edge-Antimagic Graphs AWealth

of Problems and Some Solutions Brown Walker Press BocaRaton Fla USA 2008

[4] J A Gallian ldquoDynamic survey of graph labelingrdquoTheElectronicJournal of Combinatorics vol 18 2011

[5] H Enomoto A S Llado T Nakamigawa and G Ringel ldquoSuperedge-magic graphsrdquo SUT Journal of Mathematics vol 34 no 2pp 105ndash109 1998

[6] R M Figueroa-Centeno R Ichishima and F A Muntaner-Batle ldquoThe place of super edge-magic labelings among otherclasses of labelingsrdquo Discrete Mathematics vol 231 no 1ndash3 pp153ndash168 2001

[7] A Kotzig and A Rosa ldquoMagic valuations of finite graphsrdquoCanadian Mathematical Bulletin vol 13 pp 451ndash461 1970

[8] A Kotzig and A Rosa Magic Valuations of Complete GraphsCRM Publisher 1972

[9] R Simanjuntak F Bertault and M Miller ldquoTwo new (a d)-antimagic graph labelingsrdquo in Proceedings of the 11th Aus-tralasian Workshop on Combinatorial Algorithms pp 179ndash1892000

[10] R Bodendiek and G Walther ldquoArithmetisch antimagischegraphenrdquo in Graphentheorie III K Wagner and R BodendiekEds BI-Wiss Mannheim Germany 1993

[11] N Hartsfield and G Ringel Pearls in Graph Theory AcademicPress Boston Mass USA 1990

[12] M Baca Y Lin M Miller and R Simanjuntak ldquoNew con-structions of magic and antimagic graph labelingsrdquo UtilitasMathematica vol 60 pp 229ndash239 2001

[13] K A Sugeng M Miller Slamin and M Baca ldquo(119886 119889)-edge-antimagic total labelings of caterpillarsrdquo in CombinatorialGeometry and Graph Theory vol 3330 pp 169ndash180 SpringerBerlin Germany 2005

[14] D R Silaban and K A Sugeng ldquoEdge antimagic total labelingon paths and unicyclesrdquo Journal of Combinatorial Mathematicsand Combinatorial Computing vol 65 pp 127ndash132 2008

[15] R M Figueroa-Centeno R Ichishima and F A Muntaner-Batle ldquoMagical coronations of graphsrdquoTheAustralasian Journalof Combinatorics vol 26 pp 199ndash208 2002

[16] M Baca Y Lin and A Semanicova ldquoNote on super antimag-icness of disconnected graphsrdquo AKCE International Journal ofGraphs and Combinatorics vol 6 no 1 pp 47ndash55 2009

[17] R M Figueroa-Centeno R Ichishima and F A Muntaner-Batle ldquoOn edge-magic labelings of certain disjoint unions ofgraphsrdquo The Australasian Journal of Combinatorics vol 32 pp225ndash242 2005

[18] M Baca F A Muntaner-Batle A Semanicova-FenovcıkovaandM K Shafiq ldquoOn super (119886 2)-edge-antimagic total labelingof disconnectedgraphsrdquo Ars Combinatoria In press

[19] M Baca Dafik M Miller and J Ryan ldquoAntimagic labeling ofdisjoint union of 119904-crownsrdquo Utilitas Mathematica vol 79 pp193ndash205 2009

Submit your manuscripts athttpwwwhindawicom

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Algebra

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Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


relation between (119886 119889)-EAT labeling and other labelingsnamely edge-magic vertex labeling and edge-magic totallabeling

In this paper we study super (119886 119889)-EAT labeling of theclass of graphs that can be obtained from a cycle by attachingsome pendant edges to different vertices of the cycle

2 Basic Properties

Let us first recall the known upper bound for the parameter119889 of super (119886 119889)-EAT labeling Assume that a graph 119866 hasa super (119886 119889)-EAT labeling 119891 119891 119881(119866) cup 119864(119866) rarr

1 2 |119881(119866)| + |119864(119866)| The minimum possible edge-weight in the labeling 119891 is at least 1 + 2 + (|119881(119866)| + 1) =|119881(119866)| + 4 Thus 119886 ge |119881(119866)| + 4 On the other hand themaximum possible edge-weight is at most (|119881(119866)| minus 1) +|119881(119866)| + (|119881(119866)| + |119864(119866)|) = 3|119881(119866)| + |119864(119866)| minus 1 So

119886 + (|119864 (119866)| minus 1) 119889 le 3 |119881 (119866)| + |119864 (119866)| minus 1 (1)

119889 le2 |119881 (119866)| + |119864 (119866)| minus 5

|119864 (119866)| minus 1 (2)

Thus we have the upper bound for the difference 119889 Inparticular from (2) it follows that for any connected graphwhere |119881(119866)| minus 1 le |119864(119866)| the feasible value 119889 is no morethan 3

The next proposition proved by Baca et al [12] gives amethod on how to extend an edge-antimagic vertex labelingto a super edge-antimagic total labeling

Proposition 1 (see [12]) If a graph 119866 has an (119886 119889)-EAVlabeling then

(i) 119866 has a super (119886+ |119881(119866)| +1 119889+1)-EAT labeling and

(ii) 119866 has a super (119886+ |119881(119866)|+ |119864(119866)| 119889minus1)-EAT labeling

Thefollowing lemmawill be useful to obtain a super edge-antimagic total labeling

Lemma 2 (see [13]) Let A be a sequence A = 119888 119888 + 1 119888 +

2 119888 + 119896 119896 even Then there exists a permutation Π(A) ofthe elements ofA such thatA +Π(A) = 2119888 + 1198962 2119888 + 1198962 +1 2119888 + 1198962 + 2 2119888 + 31198962 minus 1 2119888 + 31198962

Using this lemma we obtain that if 119866 is an (119886 1)-EAVgraph with odd number of edges then 119866 is also super (1198861015840 1)-EAT

3 Crowns

If 119866 has order 119899 the corona of 119866 with119867 denoted by 119866 ⊙ 119867is the graph obtained by taking one copy of 119866 and 119899 copies of119867 and joining the 119894th vertex of119866with an edge to every vertexin the 119894th copy of 119867 A cycle of order 119899 with an 119898 pendantedges attached at each vertex that is 119862119899 ⊙ 1198981198701 is called an119898-crown with cycle of order 119899 A 1-crown or only crown isa cycle with exactly one pendant edge attached at each vertex

of the cycle For the sake of brevity we will refer to the crownwith cycle of order 119899 simply as the crown if its cycle order isclear from the context Note that a crown is also known in theliterature as a sun graph In this section we will deal with thegraphs related to 1-crown with cycle of order 119899

Silaban and Sugeng [14] showed that if the 119898-crown119862119899 ⊙1198981198701 is (119886 119889)-EAT then 119889 le 5 They also describe (119886 119889)-EAT labeling of the 119898-crown for 119889 = 2 and 119889 = 4 Note thatthe (119886 2)-EAT labeling of119898-crown presented in [14] is super(119886 2)-EAT Moreover they proved that if 119898 equiv 1(mod 4) 119899and 119889 are odd there is no (119886 119889)-EAT labeling for 119862119899 ⊙ 1198981198701They also proposed the following open problem

Open Problem 1 (see [14]) Find if there is an (119886 119889)-EATlabeling 119889 isin 1 3 5 for119898-crown graphs 119862119899 ⊙ 1198981198701

Figueroa-Centeno et al [15] proved that the 119898-crowngraph has a super (119886 0)-EAT labeling

Proposition 3 (see [15]) For every two integers 119899 ge 3 and119898 ge 1 the119898-crown 119862119899 ⊙ 1198981198701 is super (119886 0)-EAT

According to inequality (2) we have that if the crown119862119899 ⊙ 1198701 is super (119886 119889)-EAT then 119889 le 2 Immediately fromProposition 3 and the results proved in [14] we have that thecrown 119862119899 ⊙ 1198701 is super (119886 0)-EAT and super (119886 2)-EAT forevery positive integer 119899 ge 3 Moreover for 119899 odd the crownis not (119886 1)-EAT In the following theorem we prove that thecrown is super (119886 1)-EAT for 119899 even Thus we partially givean answer to Open Problem 1

Theorem 4 For every even positive integer 119899 119899 ge 4 the crown119862119899 ⊙ 1198701 is super (119886 1)-EAT

Proof Let 119881(119862119899 ⊙ 1198701) = V1 V2 V119899 1199061 1199062 119906119899be the vertex set and 119864(119862119899 ⊙ 1198701) =

V1V2 V2V3 V119899V1 V11199061 V21199062 V119899119906119899 be the edge setof 119862119899 ⊙ 1198701 see Figure 1

We define a labeling 119891 119891 119881(119862119899 ⊙ 1198701) cup 119864(119862119899 ⊙ 1198701) rarr1 2 4119899 as follows

119891 (V119894) =

119894 + 119899 for 119894 = 1 2 1198992

119894 for 119894 = 1198992+ 1

119899

2+ 2 119899

119891 (119906119894) =

119894 for 119894 = 1 2 1198992

119894 + 119899 for 119894 = 1198992+ 1

119899

2+ 2 119899

119891 (V119894V119894+1) =

7119899

2+ 1 minus 119894 for 119894 = 1 2 119899

2

9119899

2+ 1 minus 119894 for 119894 = 119899

2+ 1

119899

2+ 2 119899 minus 1

119891 (V1V119899) =7119899

2+ 1

119891 (V119894119906119894) = 3119899 + 1 minus 119894 for 119894 = 1 2 119899(3)


u1

un

u2

u3

u4

unminus1

1

n

nminus1

2

3

4

⋱

Figure 1 The crown 119862119899⊙ 1198701


119908119891 (V1V119899) =11119899

2+ 2

119908119891 (V119894V119894+1) =

11119899

2+ 2 + 119894 for 119894 = 1 2 119899

2minus 1

9119899

2+ 2 + 119894 for 119894 = 119899

2119899

2+ 1 119899 minus 1

119908119891 (V119894119906119894) = 4119899 + 1 + 119894 for 119894 = 1 2 119899(4)






















(5)


119908119892 (V1V119899) = 1 + 119899

119908119892 (V119894V119894+1) = 2119894 + 119899 for 119894 = 1 2 119899 minus 1

119908119892 (V119894119906119894) = 2119894 + 119899 minus 1 for 119894 = 2 3 119899

(6)












119892 (V119894) =

119894 + 1

2 for 119894 = 1 3 119899

119899 + 119894 + 1

2 for 119894 = 2 4 119899 minus 1

119892 (119906119894) =

3119899 minus 4 + 119894

2 for 119894 = 3 5 119899

2119899 + 119894

2 for 119894 = 2 4 119899 minus 3

(7)

1

3

2

75

6

410

8

9

4

57

6

2

5

3

8

1

4

9

10

7

11

614

13

15

12

10 7

8

119

5

6

3

7

4

11

1

5

2

6

10

14

9

15

8

12

1317

18

20

16

13

14

1910

11

15

126

7

8

9













119890119894 = V119894V119896+119898+119894 119894 = 1 2 119896 (8)

and moreover

V1198962+1 V1198962+2 V119896 V119896+119898+1 V119896+119898+2 V31198962+119898 sub 119881 (119867)

(9)



119895=1119864(119865119895) cup 1198901 1198902 119890119896






119891 (V119894) = 119894 for 119894 = 1 2 2119896 + 119898

119891 (119890119894) = 3119896 + 119898 + 1 minus 119894 for 119894 = 1 2 119896

119891 (119890out119895(V119905)) =

7119896

2+ 119898 + 1 minus 119891 (V119905) + 119895 (119896 + 119898)

for 119905 = 1198962+ 1

119896

2+ 2

3119896

2+ 119898 119895 = 1 2 119903

(10)



119908119891 (119890119894) = 119891 (V119894) + 119891 (V119896+119898+1) + 119891 (119890119894)

= 119894 + (119896 + 119898 + 119894) + (3119896 + 119898 + 1 minus 119894)

= 4119896 + 2119898 + 1 + 119894

(11)


4119896 + 2119898 + 2 4119896 + 2119898 + 3 5119896 + 2119898 + 1 (12)




119908119891 (119890out119895(V119905)) = 119891 (V119905) + 119891 (V119904) + 119891 (119890

out119895(V119905))

= 119891 (V119905) + 119891 (V119904)

+ (7119896

2+ 119898 + 1 minus 119891 (V119905) + 119895 (119896 + 119898))

=7119896

2+ 119898 + 1 + 119895 (119896 + 119898) + 119891 (V119904)

(13)


4119896 + 119898 + 2 + 119895 (119896 + 119898) 4119896 + 119898 + 3 + 119895 (119896 + 119898)

5119896 + 2119898 + 1 + 119895 (119896 + 119898)

(14)


4119896 + 2119898 + 2 4119896 + 2119898 + 3 5119896 + 2119898 + 1 + 119903 (119896 + 119898)

(15)



Acknowledgments



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










u1

un

u2

u3

u4

unminus1

1

n

nminus1

2

3

4

⋱

Figure 1 The crown 119862119899⊙ 1198701


119908119891 (V1V119899) =11119899

2+ 2

119908119891 (V119894V119894+1) =

11119899

2+ 2 + 119894 for 119894 = 1 2 119899

2minus 1

9119899

2+ 2 + 119894 for 119894 = 119899

2119899

2+ 1 119899 minus 1

119908119891 (V119894119906119894) = 4119899 + 1 + 119894 for 119894 = 1 2 119899(4)






















(5)


119908119892 (V1V119899) = 1 + 119899

119908119892 (V119894V119894+1) = 2119894 + 119899 for 119894 = 1 2 119899 minus 1

119908119892 (V119894119906119894) = 2119894 + 119899 minus 1 for 119894 = 2 3 119899

(6)












119892 (V119894) =

119894 + 1

2 for 119894 = 1 3 119899

119899 + 119894 + 1

2 for 119894 = 2 4 119899 minus 1

119892 (119906119894) =

3119899 minus 4 + 119894

2 for 119894 = 3 5 119899

2119899 + 119894

2 for 119894 = 2 4 119899 minus 3

(7)

1

3

2

75

6

410

8

9

4

57

6

2

5

3

8

1

4

9

10

7

11

614

13

15

12

10 7

8

119

5

6

3

7

4

11

1

5

2

6

10

14

9

15

8

12

1317

18

20

16

13

14

1910

11

15

126

7

8

9













119890119894 = V119894V119896+119898+119894 119894 = 1 2 119896 (8)

and moreover

V1198962+1 V1198962+2 V119896 V119896+119898+1 V119896+119898+2 V31198962+119898 sub 119881 (119867)

(9)



119895=1119864(119865119895) cup 1198901 1198902 119890119896






119891 (V119894) = 119894 for 119894 = 1 2 2119896 + 119898

119891 (119890119894) = 3119896 + 119898 + 1 minus 119894 for 119894 = 1 2 119896

119891 (119890out119895(V119905)) =

7119896

2+ 119898 + 1 minus 119891 (V119905) + 119895 (119896 + 119898)

for 119905 = 1198962+ 1

119896

2+ 2

3119896

2+ 119898 119895 = 1 2 119903

(10)



119908119891 (119890119894) = 119891 (V119894) + 119891 (V119896+119898+1) + 119891 (119890119894)

= 119894 + (119896 + 119898 + 119894) + (3119896 + 119898 + 1 minus 119894)

= 4119896 + 2119898 + 1 + 119894

(11)


4119896 + 2119898 + 2 4119896 + 2119898 + 3 5119896 + 2119898 + 1 (12)




119908119891 (119890out119895(V119905)) = 119891 (V119905) + 119891 (V119904) + 119891 (119890

out119895(V119905))

= 119891 (V119905) + 119891 (V119904)

+ (7119896

2+ 119898 + 1 minus 119891 (V119905) + 119895 (119896 + 119898))

=7119896

2+ 119898 + 1 + 119895 (119896 + 119898) + 119891 (V119904)

(13)


4119896 + 119898 + 2 + 119895 (119896 + 119898) 4119896 + 119898 + 3 + 119895 (119896 + 119898)

5119896 + 2119898 + 1 + 119895 (119896 + 119898)

(14)


4119896 + 2119898 + 2 4119896 + 2119898 + 3 5119896 + 2119898 + 1 + 119903 (119896 + 119898)

(15)



Acknowledgments



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















119892 (V119894) =

119894 + 1

2 for 119894 = 1 3 119899

119899 + 119894 + 1

2 for 119894 = 2 4 119899 minus 1

119892 (119906119894) =

3119899 minus 4 + 119894

2 for 119894 = 3 5 119899

2119899 + 119894

2 for 119894 = 2 4 119899 minus 3

(7)

1

3

2

75

6

410

8

9

4

57

6

2

5

3

8

1

4

9

10

7

11

614

13

15

12

10 7

8

119

5

6

3

7

4

11

1

5

2

6

10

14

9

15

8

12

1317

18

20

16

13

14

1910

11

15

126

7

8

9













119890119894 = V119894V119896+119898+119894 119894 = 1 2 119896 (8)

and moreover

V1198962+1 V1198962+2 V119896 V119896+119898+1 V119896+119898+2 V31198962+119898 sub 119881 (119867)

(9)



119895=1119864(119865119895) cup 1198901 1198902 119890119896






119891 (V119894) = 119894 for 119894 = 1 2 2119896 + 119898

119891 (119890119894) = 3119896 + 119898 + 1 minus 119894 for 119894 = 1 2 119896

119891 (119890out119895(V119905)) =

7119896

2+ 119898 + 1 minus 119891 (V119905) + 119895 (119896 + 119898)

for 119905 = 1198962+ 1

119896

2+ 2

3119896

2+ 119898 119895 = 1 2 119903

(10)



119908119891 (119890119894) = 119891 (V119894) + 119891 (V119896+119898+1) + 119891 (119890119894)

= 119894 + (119896 + 119898 + 119894) + (3119896 + 119898 + 1 minus 119894)

= 4119896 + 2119898 + 1 + 119894

(11)


4119896 + 2119898 + 2 4119896 + 2119898 + 3 5119896 + 2119898 + 1 (12)




119908119891 (119890out119895(V119905)) = 119891 (V119905) + 119891 (V119904) + 119891 (119890

out119895(V119905))

= 119891 (V119905) + 119891 (V119904)

+ (7119896

2+ 119898 + 1 minus 119891 (V119905) + 119895 (119896 + 119898))

=7119896

2+ 119898 + 1 + 119895 (119896 + 119898) + 119891 (V119904)

(13)


4119896 + 119898 + 2 + 119895 (119896 + 119898) 4119896 + 119898 + 3 + 119895 (119896 + 119898)

5119896 + 2119898 + 1 + 119895 (119896 + 119898)

(14)


4119896 + 2119898 + 2 4119896 + 2119898 + 3 5119896 + 2119898 + 1 + 119903 (119896 + 119898)

(15)



Acknowledgments



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119890119894 = V119894V119896+119898+119894 119894 = 1 2 119896 (8)

and moreover

V1198962+1 V1198962+2 V119896 V119896+119898+1 V119896+119898+2 V31198962+119898 sub 119881 (119867)

(9)



119895=1119864(119865119895) cup 1198901 1198902 119890119896






119891 (V119894) = 119894 for 119894 = 1 2 2119896 + 119898

119891 (119890119894) = 3119896 + 119898 + 1 minus 119894 for 119894 = 1 2 119896

119891 (119890out119895(V119905)) =

7119896

2+ 119898 + 1 minus 119891 (V119905) + 119895 (119896 + 119898)

for 119905 = 1198962+ 1

119896

2+ 2

3119896

2+ 119898 119895 = 1 2 119903

(10)



119908119891 (119890119894) = 119891 (V119894) + 119891 (V119896+119898+1) + 119891 (119890119894)

= 119894 + (119896 + 119898 + 119894) + (3119896 + 119898 + 1 minus 119894)

= 4119896 + 2119898 + 1 + 119894

(11)


4119896 + 2119898 + 2 4119896 + 2119898 + 3 5119896 + 2119898 + 1 (12)




119908119891 (119890out119895(V119905)) = 119891 (V119905) + 119891 (V119904) + 119891 (119890

out119895(V119905))

= 119891 (V119905) + 119891 (V119904)

+ (7119896

2+ 119898 + 1 minus 119891 (V119905) + 119895 (119896 + 119898))

=7119896

2+ 119898 + 1 + 119895 (119896 + 119898) + 119891 (V119904)

(13)


4119896 + 119898 + 2 + 119895 (119896 + 119898) 4119896 + 119898 + 3 + 119895 (119896 + 119898)

5119896 + 2119898 + 1 + 119895 (119896 + 119898)

(14)


4119896 + 2119898 + 2 4119896 + 2119898 + 3 5119896 + 2119898 + 1 + 119903 (119896 + 119898)

(15)



Acknowledgments



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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downloads.hindawi.comdownloads.hindawi.com/journals/jam/2013/896815.pdf · On Super (,) -Edge-Antimagic Total Labeling of Special Types of Crown Graphs ... form an arithmetic progression