On a Sufficient and Necessary Condition for Graph Coloring

Transcription

1 Ope Joural of Dscrete Matheatcs, 04, 4, -5 Publshed Ole Jauary 04 ( O a Suffcet ad Necessary Codto for raph Colorg Maodog Ye Departet of Matheatcs, Zhejag Uversty, Hagzhou, Cha Eal: yd@csszjueduc eceved ugust 5, 03; revsed Septeber 3, 03; accepted October, 03 Copyrght 04 Maodog Ye Ths s a ope access artcle dstrbuted uder the Creatve Coos ttrbuto Lcese, whch perts urestrcted use, dstrbuto, ad reproducto ay edu, provded the orgal wor s properly cted I accordace of the Creatve Coos ttrbuto Lcese all Copyrghts 04 are reserved for SCIP ad the ower of the tellectual property Maodog Ye ll Copyrght 04 are guarded by law ad by SCIP as a guarda BSTCT Usg the lear space over the bary feld that related to a graph, a suffcet ad ecessary codto for the chroatc uber of s obtaed KEYWODS Vertex Colorg; Chroatc Nuber; Outer-Kerel Subspace; Plae raph Itroducto Let ( V, = be a graph, where V s a set of vertces ad E s a set of edges of vertex colorg of a graph s a colorg to all the vertces of wth p colors so that o two adjacet vertces have the sae color Such the graph s called p -colorg The al uber p s called the chroatc uber of, ad s deoted by χ ( ) The so-called Four Color Proble s that for ay plae graph, χ ( ) 4 [] The colorg of a graph s a terestg proble for ay people [] Ths s aly caused by the Four Color Proble [3] I ths paper, puttg a graph to a lear space over the bary feld F ( ), we obta the suffcet ad ecessary codto for the chroatc uber of d as a applcato of above result, we gve a characterzato for a axal plae graph to be 4-colorg The Lear Space over F() Now we troduce the lear space over the feld F ( ) Frstly, the feld F ( ) cotas oly two ebers: ( ) { 0,} F =, where the addto ad ultplcato are as usual exceptg that + = 0 Let V = { a, a,, a} be the vertces, the all vectors of the lear space are fored of the sybolc expresso It has = ( ) αa, α F vectors The addto of two vectors s defed by Here, the vertces { a a a } αa + βa = ( α + β) a = = =,,, wll serve as the ost basc eleets of the lear space They wll be as a bass of the lear space For the the basc assupto s that these vertces are learly depedet

2 M D YE ccordg to the addto F ( ), for ay vector u the lear space, t has Here we deote the zero vector by 0 For a vector = u+ u = 0 u = α a, the order of the vector u s defed by u = 'a, = where the ' eas that the addto s the usual addto the teger set vector wth order s called a -order vector, ad a vector whose order s eve s called a eve-order vector We ow gve soe structures to the lear space I other words, we wat to put a graph to the lear space I the lear space, -order vectors are vertces of a graph The edge s expressed as the -order vertex, e a s the edge aa j So we have two ways to descrbe a edge: by aa j ( the usual sese), or by a ( the lear space ) I the followg, we always dscuss a graph the lear space, t eas we express edges wth the secod for ll the -order vectors the lear space are the all possble edges wth vertces { a, a,, a}, that we deote by E : For a gvg graph ( V {,,, {,, }} E = a+ a j j j =, wth vertces, the lear space, the eleets of the set E are the -order vectors of, the the edge set E of s the subset of the set E, E E We gve two exaples here ) For the set E wth all the -order vertces, the graph K = ( V, s a coplete graph, whose ay two vertces are adjacet ) For a graph = ( V, wth vertces, the copleetary set of E the set of the -order vertces of s E \ E The the copleetary graph Ĝ of the graph s ˆ = ( V, E \ We ow see the addto For a path of wth a sequece of edges a+ b, b+ b,, b + b, where the ed-pots are a ad b, the su of the edges s: ( ) ( ) ( ) a+ b + b + b + + b + b = a+ b Ths expresso dcates the relato betwee the addto the lear space ad the coectvty of a graph That s why we put the graph to the lear space Lea The su of eve-order vectors s eve-order Ths s clear by the property that a = 0 f ad oly f = j s a specal case of the Lea, we have a 0, j =,,, are the vertces of, f Lea Let ( ) the s eve j a + a + + a = 0, Defto Let be the -desoal lear space derved by the graph ( V be a set of -order vectors Deote by e =, above, ad E the lear subspace spaed by If there are o edges of E, E = φ () the s called a outer-erel subspace of d s a axal outer-erel subspace f the ra of s axa all the outer-erel subspace of Now we gve soe basc propertes of a outer-erel subspace of a graph By defto, s a subspace of Deote the set of all -order vectors of E, the ( ) (, V E( ) ) by ( ) = s a subgraph of the copleetary graph Ĝ of, here V s the -order vectors

3 appeared E( ) The subgraph ( ) Lea 3 Let g = ( H, be a coected bloc of ( ) g, the, g s a coplete graph K the ( ) M D YE 3 cossts of soe coected blocs, ad H = { a, a,, a} be the vertces of ad { } B = a + a a + a a + a spa, 3,, s the lear subspace of Ths lea eas that every coected bloc of ( ) s a coplete graph Proof Because s spaed by -order vectors, so Suppose that a E, aj + a E, for s a lear space, the ( ) ( ) g s coected bloc of ( ) a + a + a + a = a + a j j Sce, so a + a E O the other had, f a E, the a, aj H Hece all the -order vectors fored by the set of vertces H spa the lear subspace B of Thus the coected bloc g s a coplete graph g( ) Lea 4 If a+ a + + a, the s eve ad there exsts a {,, } such that a + a Proof By the defto of, s eve For s spaed by -order vectors, so a s a coected bloc K of Thus aother vertex a ( ) of K ust appear a+ a + + a 3 The Ma esults The outer-erel subspace plays a portat role the proble of vertex colorg Theore Let be a graph wth vertces, the the suffcet ad ecessary codto for p-colorg s that the ra of a outer-erel subspace of s p Proof Frst we prove the ecessty Suppose that the graph s p -colorg The all the vertces of ca be dvded to p subsets by the colors:,,,, +,, p, to be S S S Q Q p () That eas the vertces wth a sae color are the sae subset Because t ay have a subset wth oly oe vertex, we deote the oe-vertex subsets wth dfferet colors by Q,, + Q p S =,,, are ot less the Deote the by The eleets of subset ( ) the by () ad Let the the vectors of { } S = a, a,, a, t, t t + p = (3) = {,,, 3 } E = a + a a + a a + a, =,,,, t = E E E, are depedet Hece by (3), the deso of subspace wth codto (), ad the d- It s clear that E = φ For the suffcecy, suppose that there exsts a outer- erel subspace eso of s p We dvde the vertces of to soe subsets accordg to the subspace there have = spaed by s d = t = p (4) If for two vertces a ad b a+ b, (5) the we put a ad b to a sae subset Le the otato of cogruece we deote ( od ) a b

4 4 M D YE Obvously, f a vertex a appears, the there has at least aother vertex the sae subset wth a If a vertex does ot appear, the ths vertex fors a subset by tself, e the subset cotas oly oe vertex Lea 5 The vertces fro dfferet subset are lear depedece o, e f a, a,, a belog to dfferet subsets respectvely, the ( ) a+ a + + a / 0 od I fact, f a+ a + + a a,, a a such that a + a That eas a, a s the sae subset Now we go o wth the proof of the suffcecy Suppose that the -order vectors r, r,, r p for a bass of, ad the vertces of the graph are ow dvded to the dsjot subset N, N,, Nl by the ethod above Tae b N, =,,, l, the ay vertex a of ust be soe subset N ad by (5) we have, by Lea 4, there exsts a vertex { } a = b + r, r So ay vertex of ca be expressed by b ad a eleet of b, b,, b, r, r,, r l p Thus by Lea 5, are the bass of lear space Hece l = p By the defto of ad (5) we ow that the two vertces the sae subset N are o-adjacet Thus, we ca assg oe color to the vertces of each subset N So we just eed p colors for The graph s p -colorg Due to Theore ad the expresso (4), we have: Theore For a graph wth vertces, the suffcet ad ecessary codto for χ ( ) = p s that the ra of a axal outer-erel subspace s p 4 pplcato to Plae raphs s a applcato of Theore, we cosder a result of the colorg to the plae graph axal plae graph s a graph such that for ay two o-adjacet vertces a ad b of, added to the edge ab aes a o-plaar graph It s clear that all the faces of a axal plae graph are tragles axal plae graph s 3-C-edge colorg f we ca color ts edges by 3 colors such that the three edges of every ts tragle face are colorg by dfferet colors Later we wll see that the C the defto s borrowed fro the Cauchy-ea codto the coplex fucto theory Theore 3 If a axal plae graph s 3-C-edge colorg, the the graph s 4-vertex colorg The verse of the Theore 3 s true, too That eas f a axal plae graph s 4-vertex colorg, the the graph s 3-C-edge colorg Proof We troduce the -deesoal lear space : Let ( V, ( ) ( ) ( ) {( ) ( ) ( ) ( )} = 0,0,0,,,0,, = be 3-C-edge colorg, ad the all edges of ca ap to the three eleets 0,,,0,, of by ther colors, respectvely That s the appg f such that f abc,, are the vertces of a face of, the f : E f ( a+ b) + f ( b+ c) + f ( c+ a) = 0 (6) For a path of wth the ed-pot a, b ad the sequece of the edges a+ a, a + a,, a + b, we defe ( + ) = ( + ) + ( + ) + + ( + ) f a b f a a f a a f a b (7) By the codto of 3-C-edge colorg, the extedg appg f by (7) s depedet oly o the ed-pot a ad b, ad depedet o ther path Let be the ( ) -desoal lear subspace spaed by all the -order vectors of the space The f s the hooorphc appg fro subspace oto the space The hooorphc erel cossts of such vector e of that satsfes

5 M D YE 5 Suppose s the subset of -order vectors of have f ( e ) = 0 (8) that satsfes (8) ad s spaed by The by (8) we E = φ Deote the lear depedet spag eleets of by e, e,, e, that s just the bass of d d = We tae e, e, e E such that α β γ f ( e ) ( 0, ), f ( e ) (, 0 ), f ( e ) (,) The the lear subspace = = = α β γ s spag by eα, eβ, eγ, e, e,, e Hece + 3, ad d = 4 By Theore, the graph s 4-vertex colorg [] F Harary, raph Theory, ddso-wesley, Bosto, 969 EFEENCES [] P Erdös ad Hajal, Chroatc Nuber of Fte ad Ifte raphs ad Hypergraphs, Dscrete Matheatcs, Vol 53, 985, pp 8-85 [3] N Bggs, E Lloyd ad Wlso, raph Theory , Oxford Uversty Press, Oxford, 986