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Equitable Coloring of Prism Graph and It S Central,Middle, Total And INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 8, ISSUE 08, AUGUST 2019 ISSN 2277-8616 Equitable Coloring Of Prism Graph And It’s Central,Middle, Total And Line Graph K.Praveena, M.Venkatachalam, A.Rohini, Dafik Abstract— A proper vertex coloring of a graph is equitable if the sizes of color classes differ by atmost one. The notion of equitable coloring was introduced by Meyer in 1973. In this paper we find the equitable chromatic number for Prism graph(Yn), the central graph of prism graph C(Yn), the middle graph of the prism graph M(Yn), the total graph of the prism graph T(Yn) and the line graph of the prism graph L(Yn). Index Terms—central graph, equitable coloring, equitable chromatic number, line graph, middle graph, prism graph, total graph. ———————————————————— 1 INTRODUCTION All graphs considered in this paper are connected, finite line graph of the prism graph휒 (L(푌 )). and simple. i.e.,undirected, loopless and without multiple edges. 2.1 Preliminaries: If the set of vertices of a graph G can be partitioned into k Some basic definitions are stated below for the work, classes 푉 , 푉 , … , 푉 such that each Vi is independent set and the condition ||푉 | − |푉 || ≤ 1 holds for every pair of (푖, 푗), then G Definition: 2.1 is said to be equitably k-colorable. The smallest integer k for For a given graph G = (V, E), we do an operation on G by which G is equitably k-colorable is known as Equitable subdividing each edge exactly once and joining all the non- chromatic number of G and it is denoted by 휒 (퐺) [7, 5, 10, 1]. adjacent vertices of G. The graph obtained by this process is Since Equitable coloring is a proper coloring with additional called central graph [6] of G denoted by C(G). constraints, we have 휒(퐺) ≤ 휒 (퐺) for any graph G [3]. In some discrete industrial systems one can encounter the Definition: 2.2 problem of equitable partitioning of a system with binary Let G be a graph with vertex set V (G) and the edge set conflicting relations into conflict-free sub systems. Such E(G). The middle graph [9, 10, 11] of the graph G denoted by situations can be modelled by means of a equitable graph M(G) is defined as follows: The vertex set of M(G) is V(G) ∪ coloring. E(G) in which two vertices x and y are adjacent in M(G) if the For example, in garbage collection problem [8] the vertices of following conditions holds the graph represent the garbage collection routes and pair of vertices is joined by an edge if the corresponding routes • x,y ∈ E(G), x,y are adjacent in G. should not be run on the same day. The problem of assigning • x ∈ V(G), y ∈ E(G) and they are incident in G. one of the six days of the week to each route thus reduces to the problem of six-coloring of the graph. In practice it might Definition: 2.3 be desirable to have an approximately equal number of routes Let G be a graph with vertex set V (G) and the edge set run on each of the six days. So one have to color the graph in E(G). The total graph [9, 10, 11] of the graph G is denoted by an equitable way with six colors. In contrast to the ordinary T(G) and is defined as follows: The vertex set of T(G) is V (G) ∪ proper coloring of the graph, the equitable coloring does not E(G). Two vertices x and y are adjacent in T(G) if the following possess monotonicity, namely a graph could be equitably k- conditions holds colorable without being equitably (푘 + 1) colorable. It seems to be that maximum degree plays a crucial role here [2]. • x,y ∈ E(G), x,y are adjacent in G. In this paper we investigate the equitable chromatic • x,y ∈ V (G), x,y are adjacent in G. number of Prism Graph (휒 (푌 )), the central graph of prism • x ∈V (G), y ∈ E(G), x,y are adjacent in G. graph 휒 (C(푌 )) the middle graph of the prism graph 휒 (M(푌 )), the total graph of the prism graph 휒 (T(푌 ) and the Definition: 2.4 The line graph [9, 10] of a graph G, denoted by L(G) is a ———————————————— graph whose vertices are the edges of G and if u,v ∈ E(G) then K.Praveena, Department of Computer Science, Dr.G.R.Damodaran college of uv ∈ E(L(G)) if u and v share a vertex in G. Science,Coimbatore,Tamilnadu, India. E-mail: [email protected] M.Venkatachalam Department of Mathematics, Kongunadu Arts and Science Definition: 2.5 College, Coimbatore, Tamil Nadu, India. E- mail: [email protected] A Prism graph 푌 sometimes also called a circular ladder Rohini, Department of Mathematics, Kongunadu Arts and Science College, graph and denoted by 퐶퐿 is a graph corresponding to the Coimbatore, Tamil Nadu, India. E- mail: [email protected] skeleton of an n-prism. An n-prism has 2푛 nodes and Dafik,University of Jemper, CGANT-Research Group, Department of 3푛 edges. 푌 is isomorphic to the graph cartesian product [4] Mathematics Education,Jember 68121, Indonesia. E-mail: [email protected] 퐿푛 = 퐾2 ◦ 퐶푛 where 퐾 is the complete graph with 2 vertices and Cn is the cycle of 푛 nodes. The prism graph 푌 consists an 706 IJSTR©2019 www.ijstr.org INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 8, ISSUE 08, AUGUST 2019 ISSN 2277-8616 inner cycle and an outer cycle both connected by joining the V3 = {v3i : 1 ≤ i≤ k − 1} ∪ {u3i−2 : 1 ≤ i≤ k}. corresponding vertices. Clearly V1, V2 and V3 are independent sets of V (Yn). Throughout this paper, let *푣 : 1 ≤ 푖 ≤ 푛+ and *푢 : 1 ≤ Also |V1| = 2k − 1,|V2| = |V3| = 2k. This holds the 푖 ≤ 푛+ denote the vertices of inner and outer cycle taken in inequality ||Vi| − |Vj|| ≤ 1 ∀ (i,j) satisfying equitable cyclic order, respectively. Let *푒 : 1 ≤ 푖 ≤ 푛+ and {푒 : 1 ≤ coloring. 푖 ≤ 푛+ denote the edges of inner and outer cycle taken in cyclic order, respectively. Let *푠 : 1 ≤ 푖 ≤ 푛+ denote the edge 3. If n = 3k − 2, k = 3,5,7,9,11,…, then the partition of V *푢 푣 : 1 ≤ 푖 ≤ 푛+. is done as follows: 3 EQUITABLE COLORING OF PRISM GRAPH V1 = {v3i−2 : 1 ≤ i≤ k − 1} ∪ {u3i−1 : 1 ≤ i≤ k}. Theorem 3.1 V2 = {v3i−1 : 1 ≤ i≤ k} ∪ {u3i : 1 ≤ i≤ k − 1}. The Equitable chromatic number of prism graph 푌 , where 푛 is V3 = {v3i : 1 ≤ i≤ k − 1} ∪ {u3i−2 : 1 ≤ i≤ k − 1}. any positive integer is 2, 푛 푖푠 푒푣푒푛 휒 (푌 ) = { Clearly V1, V2 and V3 are independent sets of V (Yn). 3, 푛 푖푠 표푑푑 Also |V1|=|V2|= 2k−1,|V3| = 2k−2. This holds the Proof inequality ||Vi|−|Vj|| ≤ 1∀(i,j) satisfying equitable Let 푉 (푌 ) = *푣 , 푣 , … , 푣 + ∪ *푢 , 푢 , … , 푢 + where 푣 ′s are the coloring. vertices of the inner cycle taken in cyclic order and 푢 ′s are the From case 2, χ=(Yn) ≤ 3. vertices of the outer cycle taken in cyclic order such that each Since χ(Yn) ≥ 3 and 휒 (Yn) ≥ χ(Yn) ≥ 3, we have 휒 (Yn) ≥ 푣 푢 is the edge connecting the two cycles. 3. Therefore 휒 (Yn) = 3. Case 1: If n is even Let us partition the vertex set of the prism graph V (Yn) 4 EQUITABLE COLORING OF CENTRAL GRAPH OF PRISM GRAPH as Theorem 4.1 The Equitable chromatic number of central graph of prism graph 푉 = *푣 , 푣 , … , 푣 + ∪ *푢 , 푢 , … , 푢 + 푌 , where 푛 is any positive integer is and 휒 (C(Yn)) = n, n ≥ 3. 푉 = *푣 , 푣 , … , 푣 + ∪ *푢 , 푢 , … , 푢 + Proof ’ Clearly V1 and V2 are two independent sets of V (Yn). Also Let V (Yn) = {v1,v2,··· ,vn} ∪ {u1,u2,··· ,un} where vi s are |V1| = |V2| = n. This holds the inequality ||Vi|−|Vj|| ≤ 1 ∀ the vertices of the inner cycle taken in cyclic order and (i,j) satisfying equitable coloring. ui’s are the vertices of the outer cycle taken in cyclic order and E(Yn) = {ei : 1 ≤ i ≤ n − 1} ∪ {en} ∪ {ei’ : 1 ≤ i ≤ n ⇒휒 (Yn) ≤ 2 − 1} ∪ {푒푛′} ∪ {si’ : 1 ≤ i ≤ n} where ei is the edge 푣푖 푣푖+1 (1 Since χ(Yn) ≥ 2, 휒 (Yn) ≥ χ(Yn) ≥ 2, we have 휒 (Yn) ≥ 2. ≤ i ≤ n − 1), en is the edge 푣푛푢1, 푒푖′ is the edge 푢푖 푢푖+1 (1 ≤ Therefore 휒 (Yn) = 2. i ≤ n − 1), 푒푛′ is the edge unu1 and si is the edge 푣푖 푢푖 (1 ≤ i ≤ n). Case 2: If n is odd The equitable coloring is done in three different ways By the definition of central graph 푉(퐶(푌 )) = 푉(푌 ) ∪ 퐸(푌 ) = *푣 ∶ 1 ≤ 푖 ≤ 푛+ ∪ *푢 ∶ 1 ≤ 푖 ≤ 1. If n = 3k, k = 1,3,5,7,9,··· then the partition of V is 푛+ ∪ *푣 : 1 ≤ 푖 ≤ n+ ∪ *푢 : 1 ≤ 푖 ≤ 푛+ ∪ *푠 : 1 ≤ 푖 ≤ done as follows: ′ 푛+, where 푣 , 푢 and 푠푖′ represents the edge 푒푖, 푒푖and 푠푖 V1 = {v3i−2 : 1 ≤ i≤ k} ∪ {u3i−1 : 1 ≤ i≤ k}. respectively and joining all the non-adjacent vertices of Yn in C(Yn). V2 = {v3i−1 : 1 ≤ i≤ k} ∪ {u3i : 1 ≤ i≤ k}. Case 1: n is even V3 = {v3i : 1 ≤ i≤ k} ∪ {u3i−2 : 1 ≤ i≤ k}.
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