On First-Fit Chromatic Number of Certain Graphs

Transcription

1 Internatonal Journal of Pure an Apple Mathematcs Volume 7 No. 9 07, -5 ISSN: (prnte verson); ISSN: 4-95 (on-lne verson) url: o: 0.7/pam.v79. Specal Issue pam.eu On Frst-Ft Chromatc Number of Certan Graphs V. Yenanarayanan an R.Sreeharan School of Humantes an Scences SASTRA Unversty, Thanavur - 640, TN, Ina e-mal: prof.yena@mal.com Abstract For a smple unrecte raph ts frst-ft chromatc number s the maxmum number of colors nvolve n a frst-ft colorn of t.. In ths paper, we nvestate the frst-ft chromatc numbers of unon of raphs, multpartte raphs an a raph whch s a tranulaton of a close surface an also for a specal class of bpartte raphs. In aton we ve a crsp escrpton on the computatonal aspects of the frst ft chromatc number an ncate the scope for further applcatons. We also rase some open problems. AMS Subect Classfcaton: 05C5 Key Wors anphrases:graph, Multpartte Graph, Tranulate Graph, Colorn, Frst-Ft Chromatc Number. Introucton For a smple unrecte raph G a proper colorn of ts vertces s an allotment of postve nteers referre as color so that any two aacent vertces have fferent colors. On them. A least number of colors use n such a colorn s the chromatc number of G an s enote by (G). Graph colorn s nvolve n practcal applcatons such as ata mnn, mae sementaton, resource allocaton, processes scheuln etc., Colorn technques are use n alorthms such as reey alorthm to erve upper bouns on the chromatc number. The frst-ft chromatc number, FF(G) of G s the reatest number of colors nvolve so that t has a reey colorn. Frst-ft chromatc number problem occurs n applcatons such as ynamc storae allocaton problem [4] an rao frequency allocaton problem. Gruny number s another name for frstft chromatc number an for the computaton of the same an also for certan upper an lower bouns an computatonal complexty one can see [-,5]. Here we probe FF of a) unon of raphs b) multpartte raphs

2 Internatonal Journal of Pure an Apple Mathematcs Specal Issue an c) a raph whch s a tranulaton of close surface ) a specal class of bpartte raphs..man Results Theorem.. Let FF(G) = a for =,...,. Let Ka enote the copes of the complete raph Ka on a vertces. Then a χ FF(K a ) χ FF G a. Proof.Snce FF(G) = a for =,...,, every G nclues Ka as nuce on ts vertces. So, Ka les as G nuce n χ FF(K a ) χ FF G. an so Moreover χ FF G a s obvous. Hence FF(Ka) = a an χ FF(K a ) χ FF G a. Theorem..FF(K(n, n,..., n)) = Proof. Let V(K(n, n,..., n)) = V, where n each set V no two amon n vertces are aacent for. Suppose V = { v V }, an the vertces of K(n, n,..., n) are orere as v, v V, v, v V V Let c( v ) = for v V, then c s a proper colorn usn colors. We arue that FFK(n, n,..., n) s, Suppose not an eem that V are frst-ft,v,..., V partton sets. As no two vertces, v are aacent n each +, we have for Vs for some Vs, s. The number of frst-ft partton sets + s reater than the number, there must exst an Vf such that Vf an Vf. For an element v * Vf, v * must be an element of some t, t, whch means that vertex v * s aacent wth the vertces n by the efnton of frst-ft colorn. However, the vertex v * an the elements of are n the same partte set Vf, they shoul be nepenent. Ths contracton mples that the frst-ft chromatc number of K(n, n,..., n) s. Theorem.. Let G be a tranulaton of a close surface. Then FF(G) = f an only f there s a bectve homomorphsm from the vertex set of G to the vertex set of complete equ-three partte raph Kn,n,n for some n. Proof. Suffcency part follows from Theorem.. For the other part as G s a tranulaton of a close surface, (G). But FF(G) = means (G) an so (G) =. Let the color partte sets. of G be V where the vertces n V are allotte color, for =,,. If G has three vertces, then there s a bectve homomorphsm from the vertex set of G to K,,, an we are throuh. Suppose that V(G) 4. Snce G s -connecte, each vertex of t has eree at least.

3 Internatonal Journal of Pure an Apple Mathematcs Specal Issue Let v be a vertex of G an let v, v,, vm be the nehbors of v lyn aroun v n cyclc orer: vv... vm v. We name t as the ln of v an esnate t as l(v). Suppose v has color, then the vertces n l(v) must be colore wth colors an alternatvely. Ths means that the party of vertces n l(v) must be even. an the number of vertces n l(v) s at least 4. So, there are at least two vertces n l(v) colore wth, an at least two vertces n l(v) colore wth ncatn that V an V. Smlarly by loon at vertex wth color, we euce V. We clam that G s Kn,n,n wth partte sets V, V, V. Suppose not an there are vv, vv wth vv E(G), an for any u V an v V,, uv E(G). Now conser a new partton V(G) =, where = {v, v}, = {v}, = {v}, 4 = V. Clearly =, for, an no two vertces n aacent. Now, each vertex n 4 s aacent wth each vertex n, an ; each vertex n s aacent wth each vertex n an ; each vertex n s aacent wth each vertex n, we erve that FF(G) = 4, a contracton. Hence for u V, all vertces n V an all vertces n V appear n l(u) alternately. So V = V. In a smlar manner by loon at vertex v V, we erve V = V. an we are one. Note: Theorem. can also be euce from one of the nown propertes about unquely - colorable raphs. A specal bpartte raph Gm,n shown n F. was ven n [9], where V(Gm,n) = {u, u,..., um} {v, v,..., vn} {x,..., xm} {y,..., yn} an E(G) = {(u, x) m, n} {(v, y) m, n} {(x, y) m, n} {(u, v) n} {(u, vm), (v, yn)}. Next we establsh that FF(Gm,n ) s. F.The bpartte raph Gm,n, where the parallel lnes ncate the on operaton. Yenanarayanan ave the above raph an solve the follown problem. For any three nteers, a, b, c wth a b c, oes there exsts a raph G wth (G) = a, (G) = b an (G) = c. Ths result lea to the follown problem. Problem: For postve nteer a, b, c, wth a b c, oes there exsts a raph G such that (G) = a, FF(G) = b, (G) = c an

4 Internatonal Journal of Pure an Apple Mathematcs Specal Issue (G) =. Here an enote respectvely achromatc number an pseuoachromatc number. Whle attemptn ths problem the follown result was foun.. Theorem.4.FF(Gm,n) =. Proof. The bpartte raph Gm,n has the bpartton (V, V), where V = {u} {u m} {y n} an V = {v} {v m} {x n}. For convenence, let U = {u m}, V = {v n}, X = {x n} an Y = {y n}. Let f be a frst-ft colorn of Gm,n. Fact. For any frst ft colorn f of Gm,n, the set V cannot nclue two stnct vertces one wth color an another wth color more than at a tme. Symmetrcally, the same s true for the set U. Informaton Technoloy (IJCSIT), Vol. 6, No., pp. 5560, 04. []CA.Chrsten, an SM.Selow, Some perfect colorn propertes of raphs, Journal of Combnatoral Theory, Seres B.,Vol. 7, pp. 4959, 979. [4]M.Chroba, an M. Slusare, On some pacn problems relate to ynamc storae allocaton, Informatquetheorque et Applcatons/Theoretcal Informatcs an Applcatons, Vol., No. 4,pp , 988. [5]Eouar Bonnet, FlorentFoucau, Eun Jun Km, an Floran Sora, Complexty of Gruny colorn an ts varants, arxv : v/[CS.DS], pp. 8, 04. Fact. It s not case that 4 colors can be allotte to the vertces of U V {u} n a frst-ft colorn. References [..U.Aamy, an T.Erlebach, Onlne colorn of ntervals wth banwth, In Klaus Jansen an Roberto Sols-Oba, etors, Frst Internatonal Worshop on Approxmaton an Onlne Alorthms, (WAOA 00), Lecture Notes n Computer Scence, Vol. 909, pp., 004. [].Al Mansour, an Mohame Salm Bouhlel, A Lnear Alorthm for the Gruny Number of A Tree, Internatonal Journal of Computer Scence & 4

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