Research Article Tree-Antimagicness of …downloads.hindawi.com/journals/mpe/2015/504251.pdfWhen isisomorphicto 2,asuper(,) - 2-antimagic total labeling is also called super (,) -edge-antimagic

Research ArticleTree-Antimagicness of Disconnected Graphs

Martin BaIa1 Zuzana Kimaacutekovaacute1 Andrea SemaniIovaacute-Fe^ovIiacutekovaacute1

and Muhammad Awais Umar2

1Department of Applied Mathematics and Informatics Technical University 042 00 Kosice Slovakia2Abdus Salam International Center of Mathematics Information Technology University Lahore Pakistan

Correspondence should be addressed to Martin Baca martinbacatukesk

Received 10 October 2014 Accepted 5 January 2015

Academic Editor Qing-WenWang

Copyright copy 2015 Martin Baca 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

A simple graph119866 admits an119867-covering if every edge in 119864(119866) belongs to a subgraph of119866 isomorphic to119867The graph119866 is said to be(119886119889)-119867-antimagic if there exists a bijection from the vertex set119881(119866) and the edge set119864(119866) onto the set of integers 1 2 |119881(119866)|+|119864(119866)| such that for all subgraphs1198671015840 of 119866 isomorphic to119867 the sum of labels of all vertices and edges belonging to1198671015840 constitutethe arithmetic progression with the initial term 119886 and the common difference 119889 119866 is said to be a super (119886 119889)-119867-antimagic if thesmallest possible labels appear on the vertices In this paper we study super tree-antimagic total labelings of disjoint union of graphs

1 Introduction

We consider finite undirected graphs without loops andmul-tiple edges The vertex and edge set of a graph 119866 are denotedby 119881(119866) and 119864(119866) respectively An edge-covering of 119866 is afamily of subgraphs 119867

1 1198672 119867

119905such that each edge of

119864(119866) belongs to at least one of the subgraphs119867119894 119894 = 1 2 119905

In this case we say that 119866 admits an (1198671 1198672 119867

119905)-(edge)

covering If every subgraph119867119894is isomorphic to given graph119867

then the graph 119866 admits an119867-covering A bijective function119891 119881(119866) cup 119864(119866) rarr 1 2 |119881(119866)| + |119864(119866)| which wecall a total labeling and the associated119867-weight of subgraph119867 is 119908119905

119891(119867) = sumVisin119881(119867) 119891(V) + sum

119890isin119864(119867)119891(119890) An (119886 119889)-119867-

antimagic total labeling of graph 119866 admitting an 119867-coveringis a total labeling with the property that for all subgraphs1198671015840 isomorphic to 119867 the 119867

1015840-weights form an arithmeticprogression 119886 119886 + 119889 119886 + 2119889 119886 + (119905 minus 1)119889 where 119886 gt 0 and119889 ge 0 are two integers and 119905 is the number of all subgraphsof 119866 isomorphic to 119867 Such a labeling is called a super if thesmallest possible labels appear on the vertices A graph thatadmits a (super) (119886 119889)-119867-antimagic total labeling is called a(super) (119886 119889)-119867-antimagic For 119889 = 0 it is called 119867-magicand119867-supermagic respectively

The 119867-(super)magic labelings were first studied byGutierrez and Llado [1]These labelings are the generalizationof the edge-magic and super edge-magic labelings that were

introduced by Kotzig and Rosa [2] and Enomoto et al [3]respectively For further information about (super) edge-magic labelings one can see [4ndash8] Gutierrez and Llado[1] proved that certain classes of connected graphs are 119867-(super)magic such as the star119870

1119899and the complete bipartite

graphs119870119899119898

are1198701ℎ-supermagic for some ℎThey also proved

that the cycle 119862119899is 119875ℎ-supermagic for any 2 le ℎ le 119899 minus 1

such that gcd(119899 ℎ(ℎ minus 1)) = 1 Llado and Moragas [9]studied the cycle-(super)magic behavior of several classesof connected graphs They proved that wheels windmillsbooks and prisms are119862

ℎ-magic for some ℎ Maryati et al [10]

and also Salman et al [11] proved that certain families of treesare path-supermagic Ngurah et al [12] proved that chainswheels triangles ladders and grids are cycle-supermagic

The (119886 119889)-119867-antimagic total labeling is introduced byInayah et al [13] In [14] the super (119886 119889)-119867-antimagiclabelings are investigated for some shackles of connectedgraph119867

When119867 is isomorphic to1198702 a super (119886 119889)-119870

2-antimagic

total labeling is also called super (119886 119889)-edge-antimagic totalThe notion of (119886 119889)-edge-antimagic total labeling was intro-duced by Simanjuntak et al in [15] as a natural extension ofthe edge-magic labeling defined by Kotzig and Rosa in [2] asmagic valuation

The (super) (119886 119889)-119867-antimagic total labeling is related toa super 119889-antimagic labeling of type (1 1 0) of a plane graph

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 504251 4 pageshttpdxdoiorg1011552015504251

2 Mathematical Problems in Engineering

A labeling of type (1 1 0) that is a total labeling of a planegraph is said to be 119889-antimagic if for every positive integer 119904the set of weights of all 119904-sided faces is 119882

119904= 119886119904 119886119904+ 119889 119886

119904+

2119889 119886119904+ (119891119904minus 1)119889 for some integers 119886

119904gt 0 and 119889 ge 0

where 119891119904is the number of the 119904-sided faces Note that there

are allowed different sets 119882119904for different 119904 The weight of a

face under a labeling of type (1 1 0) is the sum of labels ofall the edges and vertices surrounding that face If 119889 = 0then Lih [16] calls such labeling magic and describes magic(0-antimagic) labelings of type (1 1 0) for wheels friendshipgraphs and prisms In [17] Ahmad et al investigate theexistence of super 119889-antimagic labelings of type (1 1 0) fordisjoint union of plane graphs for several values of difference119889

In this paper we mainly investigate the existence of super(119886 1)-tree-antimagic total labelings for disconnected graphsWe concentrate on the following problem if graph119866 admits a(super) (119886 1)-tree-antimagic total labeling does the disjointunion of 119898 copies of the graph 119866 denoted by 119898119866 admit a(super) (119887 1)-tree-antimagic total labeling as well

2 Super (1198861)-Tree-Antimagic Total Labeling

In this section we will study the super (119886 1)-tree-anti-magicness for disconnected graphs The main result is thefollowing

Theorem 1 Let 119898 119905 ge 1 be positive integers Let 119866119894with a

(1198791

119894 1198792

119894 119879

119905

119894)-covering be a super (119886 1)-119879-antimagic graph

of order119901 and size 119902 119894 = 1 2 119898 where119879 is a tree and everytree 119879119895

119894 119895 = 1 2 119905 is isomorphic to 119879 Then the disjoint

union⋃119898

119894=1119866119894is also a super (119887 1)-119879-antimagic graph

Proof Let 119898 ge 1 and 119905 ge 1 be positive integers and let 119879 bea tree of order 119896 Let 119866

119894 119894 = 1 2 119898 be a graph with 119901

vertices and 119902 edges that admits a (1198791119894 1198792

119894 119879

119905

119894)-covering

Note that 119866119894is not necessarily isomorphic to 119866

119895for 119894 = 119895

Assume that every 119866119894 119894 = 1 2 119898 has a super (119886 1)-119879-

antimagic total labeling 119891119894 119881(119866

119894) cup 119864(119866

119894) rarr 1 2 119901 cup

119901 + 1 119901 + 2 119901 + 119902 thus

119908119905119891119894(119879119895

119894) 119895 = 1 2 119905 = 119886 119886 + 1 119886 + 119905 minus 1 (1)

is the set of the corresponding 119879119895119894-weights where every tree

119879119895

119894 119895 = 1 2 119905 is isomorphic to 119879Define the labeling 119891 for the vertices and edges of⋃119898

119894=1119866119894

in the following way

119891 (119909) = 119898 (119891119894(119909) minus 1) + 119894 if 119909 isin 119881 (119866

119894)

119898119891119894(119909) + 1 minus 119894 if 119909 isin 119864 (119866

119894)

(2)

First we will show that the vertices of ⋃119898119894=1

119866119894under the

labeling 119891 use integers from 1 up to 119901119898 that is if 119894 = 1 thenthe vertices of 119866

1successively attain values [1 119898 + 1 2119898 +

1 (119901minus1)119898+1] if 119894 = 2 then the vertices of1198662successively

assume values [2 119898 + 2 2119898 + 2 (119901 minus 1)119898 + 2] thevalues of vertices of119866

119894are equal successively to [119894 119898+ 119894 2119898+

119894 (119901 minus 1)119898 + 119894] and if 119894 = 119898 then the vertices of 119866119898

successively assume values [119898 2119898 3119898 119901119898]

Second we can see that the edges of ⋃119898119894=1

119866119894under the

labeling 119891 use integers from 119901119898 + 1 up to (119901 + 119902)119898 It meansthat if 119894 = 1 then the edges of 119866

1successively assume values

[(119901+ 1)119898 (119901+ 2)119898 (119901+ 3)119898 (119901 + 119902)119898] if 119894 = 2 then theedges of119866

2successively attain values [(119901+1)119898minus1 (119901+2)119898minus1

(119901 + 3)119898 minus 1 (119901 + 119902)119898 minus 1] the values of edges of 119866119894

are equal successively to [(119901 + 1)119898 + 1 minus 119894 (119901 + 2)119898 + 1 minus 119894(119901+3)119898+1minus119894 (119901+119902)119898+1minus119894] and if 119894 = 119898 then theedges of 119866

119898successively attain values [119901119898 + 1 (119901 + 1)119898 + 1

(119901 + 2)119898 + 1 (119901 + 119902 minus 1)119898 + 1]It is not difficult to see that the labeling 119891 is a bijective

function which assigns the integer 1 2 (119901 + 119902)119898 to thevertices and edges of⋃119898

119894=1119866119894 thus 119891 is a total labeling

For the weight of every subgraph 119879119895

119894isomorphic to the

tree 119879 under the labeling 119891 we have

119908119905119891(119879119895

119894) = sum

Visin119881(119879119895119894)

119891 (V) + sum

119890isin119864(119879119895

119894)

119891 (119890)

= sum

Visin119881(119879119895119894)

(119898 (119891119894(V) minus 1) + 119894)

+ sum

119890isin119864(119879119895

119894)

(119898119891119894(119890) + 1 minus 119894)

= 119898 sum

Visin119881(119879119895119894)

119891119894(V) minus 119898

10038161003816100381610038161003816119881 (119879119895

119894)10038161003816100381610038161003816+ 119894

10038161003816100381610038161003816119881 (119879119895

119894)10038161003816100381610038161003816

+ 119898 sum

119890isin119864(119879119895

119894)

119891119894(119890) +

10038161003816100381610038161003816119864 (119879119895

119894)10038161003816100381610038161003816minus 119894

10038161003816100381610038161003816119864 (119879119895

119894)10038161003816100381610038161003816

= 119898( sum

Visin119881(119879119895119894)

119891119894(V) + sum

119890isin119864(119879119895

119894)

119891119894(119890))

minus 11989810038161003816100381610038161003816119881 (119879119895

119894)10038161003816100381610038161003816+10038161003816100381610038161003816119864 (119879119895

119894)10038161003816100381610038161003816+ 119894

10038161003816100381610038161003816119881 (119879119895

119894)10038161003816100381610038161003816minus 119894

10038161003816100381610038161003816119864 (119879119895

119894)10038161003816100381610038161003816

= 119898119908119905119891119894(119879119895

119894) minus 119898

10038161003816100381610038161003816119881 (119879119895

119894)10038161003816100381610038161003816+10038161003816100381610038161003816119864 (119879119895

119894)10038161003816100381610038161003816

+ 11989410038161003816100381610038161003816119881 (119879119895

119894)10038161003816100381610038161003816minus 119894

10038161003816100381610038161003816119864 (119879119895

119894)10038161003816100381610038161003816

(3)

As every 119879119895119894 119894 = 1 2 119898 119895 = 1 2 119905 is isomorphic to

the tree 119879 it holds10038161003816100381610038161003816119881 (119879119895

119894)10038161003816100381610038161003816= |119881 (119879)| = 119896

10038161003816100381610038161003816119864 (119879119895

119894)10038161003816100381610038161003816= |119864 (119879)| = |119881 (119879)| minus 1 = 119896 minus 1

(4)

Thus for the 119879-weights we get

119908119905119891(119879119895

119894) = 119898119908119905

119891119894(119879119895

119894) + 119896 (1 minus 119898) minus 1 + 119894 (5)

According to (1) we get that the119879-weights in componentsare the following

if 119894 = 1 then the 1198791198951-weights in 119866

1are 119898(119886 minus 119896) + 119896

119898(119886 minus 119896 + 1) + 119896 and119898(119886 minus 119896 + 119905 minus 1) + 119896

Mathematical Problems in Engineering 3

if 119894 = 2 then the 1198791198952-weights in119866

2are119898(119886minus119896)+119896+1

119898(119886 minus 119896 + 1) + 119896 + 1 and119898(119886 minus 119896 + 119905 minus 1) + 119896 + 1


3are119898(119886minus119896)+119896+2


if 119894 = 119898 minus 1 then the 119879119895

119898minus1-weights in 119866

119898minus1

successively attain values 119898(119886 minus 119896) + 119896 + 119898 minus 2119898(119886minus119896+1)+119896+119898minus2 and119898(119886minus119896+119905minus1) +119896+119898minus2

if 119894 = 119898 then the 119879119895

119898-weights in 119866

119898successively

assume values119898(119886minus119896)+119896+119898minus1119898(119886minus119896+1)+119896+119898minus1 and119898(119886 minus 119896 + 119905 minus 1) + 119896 + 119898 minus 1

It is easy to see that the set of all 119879-weights in ⋃119898

119894=1119866119894

consists of distinct and consecutive integers

119908119905119891(119879119895

119894) 119894 = 1 2 119898 119895 = 1 2 119905

= 119898 (119886 minus 119896) + 119896119898 (119886 minus 119896)

+ 119896 + 1 119898 (119886 minus 119896 + 119905) + 119896 minus 1

(6)

Thus the graph ⋃119898

119894=1119866119894is a super (119898(119886 minus 119896) + 119896 1)-119879-

antimagic

Immediately from the previous theorem we get thatarbitrary number of copies of a super (119886 1)-119879-antimagicgraph is a super (119887 1)-119879-antimagic

Corollary 2 Let119866 be a super (119886 1)-119879-antimagic graph where119879 is a treeThen the disjoint union of arbitrary number of copiesof119866 that is119898119866119898 ge 1 also admits a super (119887 1)-119879-antimagictotal labeling

Moreover for 119898 copies of a graph 119866 which is (119886 1)-119879-antimagic but is not super we can derive the following result

Theorem 3 Let 119866 be an (119886 1)-119879-antimagic graph of order 119901and size 119902 where 119879 is a tree Then 119898119866 119898 ge 1 is also a (119887 1)-119879-antimagic graph

Proof Let 119879 be a tree of order 119896 and let 119866 be an (119886 1)-119879-antimagic graph of order 119901 and size 119902 with correspondinglabeling 119891 119881(119866) cup 119864(119866) rarr 1 2 119901 + 119902 and thecorresponding 119879-weights 119886 119886 + 1 119886 + 2 119886 + 119905 minus 1

For every vertex V in 119866 we denote by symbol V119894the

corresponding vertex in the 119894th copy of119866 in119898119866 Analogouslylet 119906119894V119894be the edge corresponding to the edge 119906V in the 119894th

copy of 119866 in119898119866For119898 ge 1 we define labeling 119892 of119898119866 as follows

119892 (V119894) = 119898 (119891 (V) minus 1) + 119894 for V isin 119881 (119866) 119894 = 1 2 119898

119892 (119906119894V119894) = 119898119891 (119906V) + 1 minus 119894 for 119906V isin 119864 (119866) 119894 = 1 2 119898

(7)

Let 119903 isin 1 2 119901 + 119902 We consider two cases

Case 1 If the number 119903 is assigned by the labeling119891 to a vertexof119866 then the corresponding vertices V

119894 119894 = 1 2 119898 in the

copies 119866119894in119898119866 will receive labels

119892 (V119894) 119892 (V

119894) = 119898 (119903 minus 1) + 119894 119894 = 1 2 119898

= 119898 (119903 minus 1) + 1119898 (119903 minus 1) + 2 119898119903

(8)

Case 2 If the number 119903 is assigned by the labeling119891 to an edgeof119866 then the corresponding edges 119906

119894V119894 119894 = 1 2 119898 in the

copies 119866119894in119898119866 will have labels

119892 (119906119894V119894) 119892 (119906

119894V119894) = 119898119903 + 1 minus 119894 119894 = 1 2 119898

= 119898119903119898119903 minus 1 119898 (119903 minus 1) + 1

(9)

We can see that the vertex labels and edge labels in119898119866 are notoverlapping and the maximum used label is 119898(119901 + 119902) Thus119892 is a total labeling

Analogously as in the proof of Theorem 1 we get that theweight of every subgraph 119879

119895

119894 119894 = 1 2 119898 119895 = 1 2 119905

under the labeling 119892 attains the value

119908119905119892(119879119895

119894) = sum

Visin119881(119879119895119894)

119892 (V) + sum

119906Visin119864(119879119895119894)

119892 (119906V)

= sum

Visin119881(119879119895119894)

(119898 (119891 (V) minus 1) + 119894)

+ sum

119906Visin119864(119879119895119894)

(119898119891 (119906V) + 1 minus 119894)

= 119898119908119905119891(119879119895

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

(10)

and the set of all 119879-weights in 119898119866 successively attainconsecutive values119898(119886minus119896)+119896119898(119886minus119896)+119896+1 119898(119886minus119896+

119905)+119896minus1 Thus the resulting labeling 119892 is a (119887 1)-119879-antimagictotal labeling

3 Disjoint Union of CertainFamilies of Graphs

In [18] the following results are proved on path-antimagicness of cycles and paths

Proposition 4 (see [18]) Let 119896 and 119899 ge 3 be positive integersThe cycle 119862

119899is a super (119886 1)-119875

119896-antimagic for every 119896 =

2 3 119899

Proposition 5 (see [18]) Let 119896 and 119899 ge 3 be positive integersThe path 119875

119899is a super (119886 1)-119875


2 3 119899

In light of Propositions 4 and 5 and Corollary 2 weimmediately obtain the following

Corollary 6 Let 119896 119898 ge 1 and 119899 ge 3 be positive integersThen 119898 copy of a cycle 119862

119899 that is the graph 119898119862

119899 is a super

(119886 1)-119875119896-antimagic for every 119896 = 2 3 119899


Corollary 7 Let 119896 119898 ge 1 and 119899 ge 3 be positive integersThen 119898 copy of a path 119875

119899 that is the graph 119898119875

119899 is a super


4 Conclusion

In this paper we have shown that the disjoint union ofmultiple copies of a (super) (119886 1)-tree-antimagic graph isalso a (super) (119887 1)-tree-antimagic It is a natural questionwhether the similar result holds also for other differencesand other 119867-antimagic graphs For further investigation wepropose the following open problems

Open Problem 1 Let 119866 be a (super) (119886 119889)-119879-antimagic graphwhere 119879 is a tree different from119870

2 For the graph119898119866 deter-

mine if there is a (super) (119886 119889)-119879-antimagic total labeling for119889 = 1 and all119898 ge 1

Open Problem 2 Let119866 be a (super) (119886 119889)-119867-antimagic graphFor the graph 119898119866 determine if there is a (super) (119886 119889)-119867-antimagic total labeling for certain values of 119889 and all119898 ge 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The work was supported by Slovak VEGA Grant 1005615

References

[1] A Gutierrez and A Llado ldquoMagic coveringsrdquo Journal of Com-binatorial Mathematics and Combinatorial Computing vol 55pp 43ndash56 2005

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

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

[4] M Baca F A Muntaner-Batle A Semanicova-FenovcıkovaandM K Shafiq ldquoOn super (a 2)-edge-antimagic total labelingof disconnected graphsrdquoArs Combinatoria vol 113 pp 129ndash1372014

[5] M Baca andMMiller Super Edge-Antimagic Graphs AWealthof Problems and Some Solutions Brown Walker Press BocaRaton Fla USA 2008

[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] 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

[8] A M Marr and W D Wallis Magic Graphs Birkhauser NewYork NY USA 2013

[9] A Llado and J Moragas ldquoCycle-magic graphsrdquo Discrete Math-ematics vol 307 no 23 pp 2925ndash2933 2007

[10] T K Maryati A N Salman E T Baskoro J Ryan and MMiller ldquoOn 119867-supermagic labelings for certain shackles andamalgamations of a connected graphrdquo Utilitas Mathematicavol 83 pp 333ndash342 2010

[11] A N M Salman A A G Ngurah and N Izzati ldquoOn (super)-edge-magic total labelings of subdivision of stars Snrdquo UtilitasMathematica vol 81 pp 275ndash284 2010

[12] A A Ngurah A N Salman and L Susilowati ldquo119867-supermagiclabelings of graphsrdquo Discrete Mathematics vol 310 no 8 pp1293ndash1300 2010

[13] N Inayah A N Salman and R Simanjuntak ldquoOn (119886 119889)-119867-antimagic coverings of graphsrdquo Journal of CombinatorialMathe-matics and Combinatorial Computing vol 71 pp 273ndash281 2009

[14] N Inayah R Simanjuntak A N Salman and K I SyuhadaldquoSuper (a d)-H-antimagic total labelings for shackles of aconnected graphHrdquoTheAustralasian Journal of Combinatoricsvol 57 pp 127ndash138 2013

[15] R Simanjuntak M Miller and F Bertault ldquoTwo new (a d)-antimagic graph labelingsrdquo in Proceedings of the 11th Aus-tralasian Workshop of Combinatorial Algorithm (AWOCA rsquo00)pp 179ndash189 Hunter Valley Australia July 2000

[16] KW Lih ldquoOnmagic and consecutive labelings of plane graphsrdquoUtilitas Mathematica vol 24 pp 165ndash197 1983

[17] A Ahmad M Baca M Lascsakova and A Semanicova-Fenovcıkova ldquoSuper magic and antimagic labelings of disjointunion of plane graphsrdquo Science International vol 24 no 1 pp21ndash25 2012

[18] A Semanicova-Fenovcıkova M Baca M Lascsakova MMiller and J Ryan ldquoWheels are cycle-antimagicrdquo ElectronicNotes in Discrete Mathematics In press

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of


A labeling of type (1 1 0) that is a total labeling of a planegraph is said to be 119889-antimagic if for every positive integer 119904the set of weights of all 119904-sided faces is 119882

119904= 119886119904 119886119904+ 119889 119886

119904+

2119889 119886119904+ (119891119904minus 1)119889 for some integers 119886

119904gt 0 and 119889 ge 0

where 119891119904is the number of the 119904-sided faces Note that there

are allowed different sets 119882119904for different 119904 The weight of a

face under a labeling of type (1 1 0) is the sum of labels ofall the edges and vertices surrounding that face If 119889 = 0then Lih [16] calls such labeling magic and describes magic(0-antimagic) labelings of type (1 1 0) for wheels friendshipgraphs and prisms In [17] Ahmad et al investigate theexistence of super 119889-antimagic labelings of type (1 1 0) fordisjoint union of plane graphs for several values of difference119889

In this paper we mainly investigate the existence of super(119886 1)-tree-antimagic total labelings for disconnected graphsWe concentrate on the following problem if graph119866 admits a(super) (119886 1)-tree-antimagic total labeling does the disjointunion of 119898 copies of the graph 119866 denoted by 119898119866 admit a(super) (119887 1)-tree-antimagic total labeling as well

2 Super (1198861)-Tree-Antimagic Total Labeling

In this section we will study the super (119886 1)-tree-anti-magicness for disconnected graphs The main result is thefollowing

Theorem 1 Let 119898 119905 ge 1 be positive integers Let 119866119894with a

(1198791

119894 1198792

119894 119879

119905

119894)-covering be a super (119886 1)-119879-antimagic graph

of order119901 and size 119902 119894 = 1 2 119898 where119879 is a tree and everytree 119879119895

119894 119895 = 1 2 119905 is isomorphic to 119879 Then the disjoint

union⋃119898

119894=1119866119894is also a super (119887 1)-119879-antimagic graph

Proof Let 119898 ge 1 and 119905 ge 1 be positive integers and let 119879 bea tree of order 119896 Let 119866

119894 119894 = 1 2 119898 be a graph with 119901

vertices and 119902 edges that admits a (1198791119894 1198792

119894 119879

119905

119894)-covering

Note that 119866119894is not necessarily isomorphic to 119866

119895for 119894 = 119895

Assume that every 119866119894 119894 = 1 2 119898 has a super (119886 1)-119879-

antimagic total labeling 119891119894 119881(119866

119894) cup 119864(119866

119894) rarr 1 2 119901 cup

119901 + 1 119901 + 2 119901 + 119902 thus

119908119905119891119894(119879119895

119894) 119895 = 1 2 119905 = 119886 119886 + 1 119886 + 119905 minus 1 (1)

is the set of the corresponding 119879119895119894-weights where every tree

119879119895

119894 119895 = 1 2 119905 is isomorphic to 119879Define the labeling 119891 for the vertices and edges of⋃119898

119894=1119866119894

in the following way

119891 (119909) = 119898 (119891119894(119909) minus 1) + 119894 if 119909 isin 119881 (119866

119894)

119898119891119894(119909) + 1 minus 119894 if 119909 isin 119864 (119866

119894)

(2)

First we will show that the vertices of ⋃119898119894=1

119866119894under the

labeling 119891 use integers from 1 up to 119901119898 that is if 119894 = 1 thenthe vertices of 119866

1successively attain values [1 119898 + 1 2119898 +

1 (119901minus1)119898+1] if 119894 = 2 then the vertices of1198662successively

assume values [2 119898 + 2 2119898 + 2 (119901 minus 1)119898 + 2] thevalues of vertices of119866

119894are equal successively to [119894 119898+ 119894 2119898+

119894 (119901 minus 1)119898 + 119894] and if 119894 = 119898 then the vertices of 119866119898

successively assume values [119898 2119898 3119898 119901119898]

Second we can see that the edges of ⋃119898119894=1

119866119894under the

labeling 119891 use integers from 119901119898 + 1 up to (119901 + 119902)119898 It meansthat if 119894 = 1 then the edges of 119866

1successively assume values

[(119901+ 1)119898 (119901+ 2)119898 (119901+ 3)119898 (119901 + 119902)119898] if 119894 = 2 then theedges of119866

2successively attain values [(119901+1)119898minus1 (119901+2)119898minus1

(119901 + 3)119898 minus 1 (119901 + 119902)119898 minus 1] the values of edges of 119866119894

are equal successively to [(119901 + 1)119898 + 1 minus 119894 (119901 + 2)119898 + 1 minus 119894(119901+3)119898+1minus119894 (119901+119902)119898+1minus119894] and if 119894 = 119898 then theedges of 119866

119898successively attain values [119901119898 + 1 (119901 + 1)119898 + 1

(119901 + 2)119898 + 1 (119901 + 119902 minus 1)119898 + 1]It is not difficult to see that the labeling 119891 is a bijective

function which assigns the integer 1 2 (119901 + 119902)119898 to thevertices and edges of⋃119898

119894=1119866119894 thus 119891 is a total labeling

For the weight of every subgraph 119879119895

119894isomorphic to the

tree 119879 under the labeling 119891 we have

119908119905119891(119879119895

119894) = sum

Visin119881(119879119895119894)

119891 (V) + sum

119890isin119864(119879119895

119894)

119891 (119890)

= sum

Visin119881(119879119895119894)

(119898 (119891119894(V) minus 1) + 119894)

+ sum

119890isin119864(119879119895

119894)

(119898119891119894(119890) + 1 minus 119894)

= 119898 sum

Visin119881(119879119895119894)

119891119894(V) minus 119898

10038161003816100381610038161003816119881 (119879119895

119894)10038161003816100381610038161003816+ 119894

10038161003816100381610038161003816119881 (119879119895

119894)10038161003816100381610038161003816

+ 119898 sum

119890isin119864(119879119895

119894)

119891119894(119890) +

10038161003816100381610038161003816119864 (119879119895

119894)10038161003816100381610038161003816minus 119894

10038161003816100381610038161003816119864 (119879119895

119894)10038161003816100381610038161003816

= 119898( sum

Visin119881(119879119895119894)

119891119894(V) + sum

119890isin119864(119879119895

119894)

119891119894(119890))

minus 11989810038161003816100381610038161003816119881 (119879119895

119894)10038161003816100381610038161003816+10038161003816100381610038161003816119864 (119879119895

119894)10038161003816100381610038161003816+ 119894

10038161003816100381610038161003816119881 (119879119895

119894)10038161003816100381610038161003816minus 119894

10038161003816100381610038161003816119864 (119879119895

119894)10038161003816100381610038161003816

= 119898119908119905119891119894(119879119895

119894) minus 119898

10038161003816100381610038161003816119881 (119879119895

119894)10038161003816100381610038161003816+10038161003816100381610038161003816119864 (119879119895

119894)10038161003816100381610038161003816

+ 11989410038161003816100381610038161003816119881 (119879119895

119894)10038161003816100381610038161003816minus 119894

10038161003816100381610038161003816119864 (119879119895

119894)10038161003816100381610038161003816

(3)

As every 119879119895119894 119894 = 1 2 119898 119895 = 1 2 119905 is isomorphic to

the tree 119879 it holds10038161003816100381610038161003816119881 (119879119895

119894)10038161003816100381610038161003816= |119881 (119879)| = 119896

10038161003816100381610038161003816119864 (119879119895

119894)10038161003816100381610038161003816= |119864 (119879)| = |119881 (119879)| minus 1 = 119896 minus 1

(4)

Thus for the 119879-weights we get

119908119905119891(119879119895

119894) = 119898119908119905

119891119894(119879119895

119894) + 119896 (1 minus 119898) minus 1 + 119894 (5)

According to (1) we get that the119879-weights in componentsare the following

if 119894 = 1 then the 1198791198951-weights in 119866

1are 119898(119886 minus 119896) + 119896

119898(119886 minus 119896 + 1) + 119896 and119898(119886 minus 119896 + 119905 minus 1) + 119896



2are119898(119886minus119896)+119896+1



3are119898(119886minus119896)+119896+2


if 119894 = 119898 minus 1 then the 119879119895


119898minus1


if 119894 = 119898 then the 119879119895

119898-weights in 119866

119898successively



119894=1119866119894


119908119905119891(119879119895

119894) 119894 = 1 2 119898 119895 = 1 2 119905

= 119898 (119886 minus 119896) + 119896119898 (119886 minus 119896)

+ 119896 + 1 119898 (119886 minus 119896 + 119905) + 119896 minus 1

(6)


119894=1119866119894is a super (119898(119886 minus 119896) + 119896 1)-119879-

antimagic











(7)



119894 119894 = 1 2 119898 in the


119892 (V119894) 119892 (V

119894) = 119898 (119903 minus 1) + 119894 119894 = 1 2 119898

= 119898 (119903 minus 1) + 1119898 (119903 minus 1) + 2 119898119903

(8)


119894V119894 119894 = 1 2 119898 in the


119892 (119906119894V119894) 119892 (119906

119894V119894) = 119898119903 + 1 minus 119894 119894 = 1 2 119898

= 119898119903119898119903 minus 1 119898 (119903 minus 1) + 1

(9)



119895

119894 119894 = 1 2 119898 119895 = 1 2 119905


119908119905119892(119879119895

119894) = sum

Visin119881(119879119895119894)

119892 (V) + sum

119906Visin119864(119879119895119894)

119892 (119906V)

= sum

Visin119881(119879119895119894)

(119898 (119891 (V) minus 1) + 119894)

+ sum

119906Visin119864(119879119895119894)

(119898119891 (119906V) + 1 minus 119894)

= 119898119908119905119891(119879119895

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

(10)






119899is a super (119886 1)-119875


2 3 119899


119899is a super (119886 1)-119875


2 3 119899



119899 that is the graph 119898119862

119899 is a super




119899 that is the graph 119898119875

119899 is a super


4 Conclusion








Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2are119898(119886minus119896)+119896+1



3are119898(119886minus119896)+119896+2


if 119894 = 119898 minus 1 then the 119879119895


119898minus1


if 119894 = 119898 then the 119879119895

119898-weights in 119866

119898successively



119894=1119866119894


119908119905119891(119879119895

119894) 119894 = 1 2 119898 119895 = 1 2 119905

= 119898 (119886 minus 119896) + 119896119898 (119886 minus 119896)

+ 119896 + 1 119898 (119886 minus 119896 + 119905) + 119896 minus 1

(6)


119894=1119866119894is a super (119898(119886 minus 119896) + 119896 1)-119879-

antimagic











(7)



119894 119894 = 1 2 119898 in the


119892 (V119894) 119892 (V

119894) = 119898 (119903 minus 1) + 119894 119894 = 1 2 119898

= 119898 (119903 minus 1) + 1119898 (119903 minus 1) + 2 119898119903

(8)


119894V119894 119894 = 1 2 119898 in the


119892 (119906119894V119894) 119892 (119906

119894V119894) = 119898119903 + 1 minus 119894 119894 = 1 2 119898

= 119898119903119898119903 minus 1 119898 (119903 minus 1) + 1

(9)



119895

119894 119894 = 1 2 119898 119895 = 1 2 119905


119908119905119892(119879119895

119894) = sum

Visin119881(119879119895119894)

119892 (V) + sum

119906Visin119864(119879119895119894)

119892 (119906V)

= sum

Visin119881(119879119895119894)

(119898 (119891 (V) minus 1) + 119894)

+ sum

119906Visin119864(119879119895119894)

(119898119891 (119906V) + 1 minus 119894)

= 119898119908119905119891(119879119895

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

(10)






119899is a super (119886 1)-119875


2 3 119899


119899is a super (119886 1)-119875


2 3 119899



119899 that is the graph 119898119862

119899 is a super




119899 that is the graph 119898119875

119899 is a super


4 Conclusion








Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899 that is the graph 119898119875

119899 is a super


4 Conclusion








Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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