T-Coloringon Certain Graphs

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Integrated Intelligent Research (IIR)

International Journal of Computing Algorithm Volume: 03 Issue: 02 June 2014 Pages: 142-143 ISSN: 2278-2397

T-Coloringon Certain Graphs Antony Xavier1,R.C. Thivyarathi 2 1

Department of Mathematics, Loyola College, Chennai Department of Science and Humanities, R.M.D Engineering College, Chennai

2

AbstractGivenagraph andasetT ofnonnegativeintegers containing0, aT coloringofGisanintegerfunction oftheverticesof Gsuchthat whenever .Theedgespanofa - c o l o r i n g i s t h e m a x i m u m v a l u e o f over all edges , and the e d g e - s p a n ofagraph G is theminimum value of the edgespanamongallpossibleT-colorings ofG.ThispaperdiscussestheTedgespanofthefolded hypercube networkofdimensionnforthekmultiple-of-sset, where s and and .

I.

INTRODUCTION

Inthechannelassignmentproblem,severaltransmittersandaforbiddensetT (calledT-set)ofnonnegativeintegerscontaining0,aregiven. Weassignanon- negative integerchannel toeachtransmitterunderaconstraint:fortwotransmitters wherepotentialinterferencemightoccur,thedifferenceoftheirchannelsdoesnot fallwithinthegiven -set. Theinterferenceisduetovariousreasonssuchas geographicproximityandmeteorologicalfactors. Toformulatethisproblem,we constructagraphGsuchthateachvertexrepresentsatransmitter,andtwovertices areadjacentifthepotentialinterferenceoftheircorrespondingtransmittersmight occur.Thus,wehavethefollowingdefinition. GivenaT-setandagraphG, a c o l o r i n g ofGisafunction suchthat tion suchthat if . Notethatif ,then coloringisthesameasordinary vertexcoloring.HencewemayconsidertheTcoloringproblemisageneralized graphvertex-coloringproblem. T-coloringproblemhasbeenstudiedbyseveral authors,suchas[1,5,6,9,10,13]and[15].Let bea coloringforagraphG. Therearethreeimportantcriteriaformeasuringtheefficiencyof :First,theorderofaTcoloring,whichisthenumber susedin ;second,thespan of ,whichisthemaximumof overallverticesuandv;andthird,theedge-spanof ,whichisthe maximumof overalledges . Given andG,the chromatic numberχT(G) istheminimumorderamongallpossible -coloringsofG,theT-spanspT(G)isthe minimumspana-

setscanbevery complexanddifficulttomodel.Wefocusonaspecialfamily -setscalledthe - multipleof-ssetswhichhastheform ,wheres,k≥1and . Thek-multiple-of-ssetsbasbeenstudiedbyRaychaudhurifirst(see[11,12]). Whens=1,theset isalsocalledanrinitialset. Somepracticalforbiddensets,suchasthosethatariseinUHFtelevisionproblem (see[14]),areverysimilartok-multiple-of-s-sets.WedenoteKn asthecompletegraph(orclique)onn verticesandω(G)asthe maximumsizeofacliqueinG. Definition -An uniform -star split graph contains a clique such that the deletion of the edges of partitions the graph into independant star graphs . See Figure 1. The number of vertices in is and the number of edges is . In the following theorem, we arrive at lower bound for , when is and . See Fig 1.

Figure:1 Theorem1:The T-coloring of a uniform n- star split graph is . Proof. First we name the vertices of the complete graph by in clockwise direction. Then start naming the vertices which are adjacent to in clockwise direction as .where

denotes the number of elements in T.

mongallpossibleT-coloringsofG,andthe -edge-span espT(G)istheminimumedge-spanamongallpossible -colorings ofG.Inthecaseofradiofrequencyassignment,theforbidden -

Fig 2 n- star split graph 142


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