SOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH.
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1 SOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH. K VASUDEVAN, K. SWATHY AND K. MANIKANDAN 1 Dpartmnt of Mathmatics, Prsidncy Collg, Chnnai-05, India. vasu k dvan@yahoo.com. 2, Dpartmnt of Mathmatics, Guru Nanak Collg, Chnnai-2, India. 2 kswathyka@gmail.com kmanimaths1987@gmail.com, Abstract Coloring th vrtics of a graph G according to crtain condition is a random xprimnt and a discrt random variabl X is dfind as th numbr of vrtics having a particular color in th givn typ of coloring of G and a probability mass function for this random variabl can b dfind accordingly. In this papr w xtnd th concpts of arithmtic man and varianc to th thory of quitabl graph coloring and dtrmin th valus of ths paramtrs. AMS Subjct Classification: 05C15, 62A01. Kywords: Graph coloring, Coloring man, Coloring varianc, χ -chromatic man, χ -chromatic varianc, χ -chromatic man, χ -chromatic varianc. 1 Introduction All graphs considrd in this papr ar finit, looplss and without multipl dgs. A graph G = (V (G), E(G)) is said to b quitably k-colorabl if th vrtx st V (G) can b partitiond into k indpndnt substs V 1, V 2,..., V K such that V i V j 1 for all i and j. Each V i is said to form a color class. Th smallst intgr k for which G is quitably k-colorabl is calld th quitabl chromatic numbr of G. W. Myr [9] introducd th notion of quitabl colorability. Equitabl Coloring Conjctur (ECC) [9] For any connctd graph G, which is nithr a complt graph nor an odd cycl,χ (G) (G), whr (G) is th maximum vrtx dgr in G. JGRMA 2018, All Rights Rsrvd 91
2 Sudv t.al [10] hav found th man and varianc of quitabl coloring of crtain graphs. K.P.Chithra t.al [5] found th quitabl coloring paramtrs of crtain whl rlatd graphs. Sudha t.al [11, 12] hav discussd th total coloring of Prism and X(Z n, C) graph. For a propr k-coloring {c 1, c 2,..., c k } of G, w can dfin a random variabl (r.v)x which dnots th color of any arbitrary vrtx in G. As th sum of all wights of colors of G is th ordr of G, th ral valud function f(i) dfind by θ(c i ) V (G), if i = 1, 2,,..., k is th probability mass function (p.m.f) of th random variabl (r.v)x Dfinition 1.1. Lt {c 1, c 2,..., c k } b a crtain typ of propr k-coloring of a givn graph G and f(i) dnots th p.m.f of a particular color c i assignd to th vrtics of G. Thn, 1. Th coloring man of a coloring of a givn graph G, dnotd by µ c (G) or simply µ(g), is dfind to b µ c (G) = E(i) = k i.f(i) and i=1 2. Th coloring varianc of is dfind as σc 2 (G) = V (i) = k (i µ) 2 f(i). 1.1 Equitabl Chromatic Paramtrs of Graphs An quitabl coloring of a graph G is a propr coloring of G with an assignmnt of colors to th vrtics of G such that th numbr of vrtics in any two color classs diffr by at most on. Th quitabl chromatic numbr of a graph G is th smallst numbr k such that G has an quitabl coloring with k colors. Dfinition 1.2. Th coloring man of a graph G with rspct to a propr coloring is said to b an quitabl chromatic man or χ -chromatic man of G if it is th minimum quitabl coloring of G and th coloring man is also minimum. Th χ -chromatic man of a graph G is dnotd by µ χ (G) Dfinition 1.. Th χ -chromatic varianc of G dnotd by σ 2 χ i=1 is a coloring varianc of G with rspct to a minimal quitabl coloring of G which yilds th minimum coloring sum. Dfinition 1.. A coloring man of a graph G, with rspct to a propr coloring is said to b a χ -chromatic man of G if it is th minimum quitabl colouring of G, but th colouring man is maximum. Th χ -chromatic man of a graph G is dnotd by µ χ (G). Dfinition 1.5. Th χ -chromatic varianc of G dnotd by σ 2 (G), is a coloring varianc of G χ with rspct to a minimal quitabl coloring of G which yilds th maximum coloring sum. JGRMA 2018, All Rights Rsrvd 92
3 Dfinition 1.6. A Prism Y n is th Cartsian product of th cycl C n and th p a t h P n. Dfinition 1.7. [6] L t Z n dnot th additiv group of intgrs modulo n. If C is a subst of Z n \ 0, thn construct a graph X = X(Z n, C) as follows. Th vrtics of X ar th lmnts of Z n and (i, j) is an arc of X if and only if j i C. Th graph X(Z n, C) is calld a circulant of ordr n, and C is calld its connction st. Sinc X(Z n, C)-graph satisfis th conditions rquird for an Hamiltonian and an Eulrian, it is both Hamiltonian and Eulrian. Hr th connction st C is {±1, ±2} Equitabl coloring is bing usd in schduling problms in which jobs hav to ballocatd to workrs such that th maximum diffrnc btwn any two workrs in th allocation is on.in this papr w find th χ -chromatic man, χ -chromatic varianc, χ -chromatic man, χ -chromatic varianc of Prism Y n and S(n, 2)-graph. 2 Th χ and χ -chromatic paramtrs of Prism Thorm 2.1. Th χ -chromatic man of th prism Y n is, if n is vn 2 µ χ (Y n ) = 2, if n is odd and n 0(mod ) n 1, if n is odd and n 1, 2(mod ) and th χ -chromatic varianc of th prism Y n is 1, if n is vn 2, if n is odd and n 0(mod ) σχ 2 (Y n ) = 8n 2 1 2, if n is odd and n 1(mod ) 8n 2 1 2, if n is odd and n 2(mod ) Proof. Lt {v i, 1 i n} bth vrtx st of Y n. Suppos n is vn. Thn Y n is bipartit and can b colord with 2-colors c 1 and c 2 which will b a minimal quitabl coloring of Y n. Thn,µ χ (Y n ) = 2 and σ2 χ (Y n ) = 1. Lt n ban oddintgr. Thn any propr coloring of Y n must contain atlast thr colors say c 1, c 2, and c. Lt C 1, C 2 and C bth rspctiv color classs. Thn w hav th following thr cass. JGRMA 2018, All Rights Rsrvd 9
4 Cas 1: n 0(mod ) Color th vrtics of Y n in such a way that C 1 = {v i : i 0(mod )}, C 2 = {v i : i 1(mod )} and C = {v i : i 2(mod )}. Thn ach color class will hav xactly n quitabl coloring of Y n with its p.m.f dfind as 1, i = 1, 2, Hnc th paramtrs ar µ χ (Y n ) = 2 and σ 2 χ (Y n ) = 2. Cas 2: n 1(mod ) vrtics and hnc is a minimal In this cas C 1 contains 1 vrtics and th rmaining two color classs contain 1 vrtics ach. Hnc th corrsponding p.m.f is dfind as, 2(n 1) n, i = 1 6n, i = 1, 2 Hnc µ χ (Y n ) = n 1 Cas : n 2(mod ) and σ2 χ (Y n ) = 8n In this cas C 1 contains 1 vrtics and th rmaining two color classs contain 2 vrtics ach. Hnc th corrsponding p.m.f is dfind as, 1 6n, i = 2, 1 6n, i = 1 Hnc µ χ (Y n ) = n 1 and σ2 χ (Y n ) = 8n Thorm 2.2. Th χ -chromatic man of th prism Y n is, if n is vn 2 µ χ (Y n ) = 2, if n is odd and n 0(mod ) n 1, if n is odd and n 1, 2(mod ) JGRMA 2018, All Rights Rsrvd 9
5 and th χ -chromatic varianc of th prism Y n is 1, if n is vn 2, if n is odd and n 0(mod ) σ 2 χ (Y n) = 8n 2 1 2, if n is odd and n 1(mod ) 8n 2 1 2, if n is odd and n 2(mod ) Proof. F o r th cass whn n is vn and n is oddwith n 0(mod ) all th coloring classs hav th sam numbr of vrtics. Hnc, rvrsing th coloring pattrn will also giv th sam man and varianc. That is if n is vn thn µ χ (Y n ) = 2 and σ2 (Y χ n ) = 1. Also whn n is oddand n 0(mod ), µ (Y n) = 2 and σ 2 χ (Y χ n ) = 2. Whn n is oddand n 1(mod ) rvrsing th coloring pattrn w hav n 1 n, i = 1 1 6n, i = 2, Hnc µ χ (Y n ) = n1 and σ2 (Y χ n ) = 8n2 1 2 Similarly by rvrsing th cloring pattrn for th cas n is oddand n 2(mod ) 1 6n, i = 1, 2 2 6n, i = Hnc µ χ (Y n ) = n1 and σ2 χ (Y n ) = 8n Th χ and χ chromatic paramtrs of th X(Z n, C) Graphs Thorm.1. Th χ -chromatic man of X(Z n, C) is 2, if n 0(mod ) 5, if n 0(mod ) µ χx(zn,c) = 2 5n, if n 1, (mod ) 5n, if n 2(mod ) JGRMA 2018, All Rights Rsrvd 95
6 and th χ -chromatic varianc of X(Z n, C) is Proof. 2, if n 0(mod ) 5, if n 0(mod ) σχ 2 5n 2 n 9 X(Z n, C) = n 2, if n 1(mod ) 5n , if n 2(mod ) 5n n 9 n 2, if n (mod ) In X(Z n, C) has th maximum dgr of a vrtx is. Hnc any propr quitabl coloring may hav at most colors. Cas 1: n 0(mod ) In this cas th graph has an quitabl coloring with thr colors and hnc ach color class will hav n vrtics. By dfining th p.m.f 1, i = 1, 2, w gt µ χ (X(Z n, C)) = 2 and σ 2 (X(Z χ n, C)) = 2 Cas 2: n 0(mod ) In this cas th vrtx st can bsplit up into four color classs ach having n th p.m.f vrtics. By dfining 1, i = 1, 2,, w gt µ χ (X(Z n, C)) = 5 2 and σ2 χ (X(Z n, C)) = 5 Cas : n 1(mod ) Th first color class will hav n 1 vrtics and th rmaining thr classs will hav n ach.thrfor th p.m.f is givn by n 1 n, i = 2,, n n, i = 1 which yilds µ χ (X(Z n, C)) = 5n and σ2 χ (X(Z n, C)) = 5n2 n 9 n 2 JGRMA 2018, All Rights Rsrvd 96
7 Cas : n 2(mod ) In this cas th two color classs will hav n 2 vrtics and th rmaining thr classs will hav n2 ach. Dfin th p.m.f as which yilds µ χ (X(Z n, C)) = 5n Cas 5: n (mod ) n 2 n, i =, n2 n, i = 1, 2 and σ2 χ (X(Z n, C)) = 5n2 16 n 2 In this cas th first color classs will hav n vrtics and th rmaining thr classs will hav n1 ach.th p.m.f is givn by n n, i =, n1 n, i = 1, 2 which yilds µ χ (X(Z n, C)) = 5n and σ2 χ (X(Z n, C)) = 5n2 n 9 n 2 Thorm.2. Th χ -chromatic man of X(Z n, C) is and th χ -chromatic varianc of X(Z n, C) is 2, if n 0(mod ) 5, if n 0(mod ) µ χ X(Z n, C) = 2 5n, if n 1, (mod ) 5n, if n 2(mod ) 2, if n 0(mod ) 5, if n 0(mod ) σ 2 χ (X(Z 5n 2 n 9 n, C)) = n 2, if n 1(mod ) 5n , if n 2(mod ) 5n n 9 n 2, if n (mod ) JGRMA 2018, All Rights Rsrvd 97
8 Proof. F o r th cass with n 0(mod ) and n 0(mod ) all th coloring classs hav th sam numbr of vrtics. Hnc rvrsing th coloring pattrn will also giv th sam man and varianc. Hnc µ χ (X(Z n, C)) = 2 and σ 2 (X(Z χ n, C)) = 2 for n 0(mod ) µ χ (X(Z n, C)) = 5 2 and σ2 (X(Z χ n, C)) = 5 for n 0(mod ) By rvrsing th coloring pattrn for th cas n 1(mod ) th pmf is n 1 n, i = 1, 2, n n, i = Hnc µ χ (X(Z n, C)) = 5n and σ2 (X(Z χ n, C)) = 5n2 n 9 n 2 F o r th cas n 2(mod ) th p.m.f is n 2 n, i = 1, 2 n2 n, i =, Hnc µ χ (X(Z n, C)) = 5n and σ2 (X(Z χ n, C)) = 5n2 16 n 2 F o r th cas n (mod ) th p.m.f is n n, i = 1 n1 n, i = 2,, Hnc µ χ (X(Z n, C)) = 5n and σ2 χ (X(Z n, C)) = 5n2 n 9 n 2 Rfrncs [1] A.Brandstadt, V.B.L and J.P.Spinard, Graph classs: A survy, SIAM, Philadlphia, [2] J.A.Bondy and U.S.R. Murty, Graph thory, Springr, Brlin, [] G.Chartrand and P.Zhang, Introduction to graph thory, McGraw-Hill Inc., Nw York,2005. [] G.Chartrand and P.Zhang, Chromatic graph thory,crc Prss, Boca Raton, [5] K.P. Chithra, E.A. Shiny and N.K.Sudv, On quitabl coloring paramtrs of crtain whl rlatd graphs, Discrt Math. Algorithm. Appl., 9(), [6] Godsil, Chris and Royl, Gordon, Algbraic Graph Thory, Springr (2009). JGRMA 2018, All Rights Rsrvd 98
9 [7] F.Harary, Graph thory, Narosa Publishing Hous, Nw Dlhi, [8] T.R. Jnsn and B.Toft, Graph colouring problms,john Wily and Sons, Nw Jrsy, [9] W.Myr, Equitabl coloring, Amr. Math. Monthly, 80, ,197. [10] S.M.Ross, Introduction to probability and statistics for Enginrs and scintists, Acadmic Prss, Massachustts, 200. [11] N.K.Sudv,K.P.Chithra, S.Sathsh and J.Kok, On crtain paramtrs of quitabl coloring of graphs, Discrt Math. Algorithm. Appl., 9(), 2017 JGRMA 2018, All Rights Rsrvd 99
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