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On Equitable Coloring Parameters of Certain Graphs Journal of Information and Computational Science ISSN: 1548-7741 ON EQUITABLE COLORING PARAMETERS OF CERTAIN GRAPHS 1. J.Mariya sowmiya, 2. Dr. Sr. A. Stanis Arul Mary 1. PG Scholar, Department of Mathematics, Nirmala College for Women, Red Fields, Coimbatore. E-Mail: [email protected] 2. Assistant Professor, Department of Mathematics, Nirmala College for Women, Red fields,Coimbatore. Abstract A proper coloring 풞 of a graph G, is assigning colors to the vertices of G such that the numbers of vertices in any two color classes differ by at most one, is called an equitable colouring of G. In this paper, we study two statistical parameters of five different graphs and we apply equitable coloring for these graphs. Here the graphs are Tadpole, windmill, friendship, gear and sunlet. Keywords: Equitable coloring; coloring mean; coloring variance; χ푒 – Chromatic mean; χ푒 – Chromatic variance, Tadpole graph(푇(푚,푛)), Windmill graph(푊푑(푘, 푛)), Friendship graph(퐹푛), Gear graph(퐺푛), Sunlet graph(푆푛). 1 Introduction In this paper all the graphs we are considering are simple, finite, connected and undirected. Graph coloring is an assignment of colors or labels or weights to the vertices, edges and faces of a graph under consideration. Many practical and real life situations motivated in the development of different types of graph coloring problems. Unless stated otherwise, the graph coloring is meant to be an assignment of colors to the vertices of a graph subject to certain conditions. A proper coloring of a graph G is a colouring with respect to which vertices of G are colored in such a way that no two adjacent vertices G have the same color. Coloring of the vertices of a given graph G can be considered as a random experiment. For a proper k-coloring 풞 = {푐1,푐2,푐3,……,푐푘} of G, we can define a random variable (r.v) X which denotes the color ( or precisely, the subscript i of the color 푐푖) of any arbitrary vertex in G. As the sum of all weights of colors of G is equal to the number of vertices of G, the real valued function 푓(i) defined by, Volume 10 Issue 1 - 2020 891 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741 휃(푐 ) 푖 ; = 1,2, … . , 푘 푓() = {|푉(퐺)| 0 ; 푒푙푠푒푤ℎ푒푟푒 will be the probability mass function (p.m.f.) of the random variable (r.v.) X, (see [7]),where 휃(푐푖) denotes the cardinality of the colour class of the colour 푐푖. If the context is clear, we can also say that 푓(i) is the probability mass function of the graph G with respect to the given coloring 풞. Hence, we can also define the parameters like mean and variance of the random variable X, with respect to a general coloring of G can be defined as follows. An equitable coloring of G is a proper coloring 풞 of G which an assignment of colors to the vertices of G such that the numbers of vertices in any two color classes differ by at most one. The equitable chromatic number of a graph G is the smallest number k such that G has an equitable coloring with k colors. Throughout the paper, we follow the convention that 0≤ 휃(푐푖) - 휃(푐푗) ≤ 1, when i < j. 2 Preliminaries Two types of chromatic means and variances corresponding to an equitable coloring of a graph G are defined as follows. [8] Let 풞 = {푐1,푐2,푐3,……,푐푘} be an equitable k-coloring of a given graph G and X be the random variable which denotes the number of vertices having a particular color in 풞, with the p.m.f 푓(). Then, (i) The equitable coloring mean of a coloring 풞 of a given graph G, denoted by 푛 휇휒푒 (G), isdefined to be to be 휇휒푒 (G) = ∑푖=1 푓(); (ii) Equitable coloring variance of a coloring 풞 of a given graph G, denoted by 2 2 푛 2 푛 2 휎휒푒(퐺), 푠 defined to be 휎휒푒(퐺) = ∑푖=1 푓()– (∑푖=1 푓()) . If the context is clear, the above defined parameters can respectively be called the equitable coloring mean and equitable coloring variance of the graph G with respect to the coloring 풞. In this paper we study the equitable coloring parameters of certain graphs such as Tadpole, Windmill, Friendship, Gear and Sunlet. Definitions: (i) Tadpole Graph [11] A tadpole graph 푇(푚,푛) is a graph obtained by joining a cycle 퐶푚, m≥ 3 to a path 푃푛. n ≥ 1 with a bridge. (ii) Windmill Graph [12] The windmill graph 푊푑(k,n) is an undirected graph constructed for k ≥ 2, n≥ 2 by joining n copies of the complete graph 퐾푘at a shared universal vertex. That is a 1-clique sum of these complete graphs. Volume 10 Issue 1 - 2020 892 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741 (iii)Friendship Graph [13] A friendship graph 퐹푛 is a planar undirected graph with 2n+1 vertices and 3n edges. It can be constructed by joining n copies of the “ cycle graph” 퐶3 with a common vertex. (iv)Gear Graph [14] A gear graph denoted 퐺푛 is a wheel graph with a graph vertex added between each pair of adjacent graph vertices of the outer cycle. Thus퐺푛 has 2n+1 vertices and 3n edges. (v)Sunlet Graph [15] The n-sunlet graph of 2n vertices is obtained by attaching n pendant edges to a cycle퐶푛and is denoted by 푆푛. Sunlet graphs are also been called as “crown graph”. Theorem 1.For a Tadpole graph 푇(3,푛), we have 푛2+2푛+4 , 푓 푛 푠 푒푣푒푛 휇 (푇 ) = { 4푛 휒푒 (3,푛) 푛2+2푛+1 , 푓 푛 푠 표푑푑 4푛 and 푛4+20푛2−48 2 , 푓 푛 푠 푒푣푒푛 휎2 (푇 ) = { 48푛 휒푒 (3,푛) 푛4+2푛2−3 , 푓 푛 푠 표푑푑 48푛2 Proof. Let푣1, 푣2, 푣3,……….푣푛 be the vertices of a tadpole graph 푇(3,푛). Here 3 represents the total number of vertices on a cycle. n denotes the number of vertices connected by a path. Hence, every other colour class, with respect to an equitable colouring of 푇(3,푛) must be a singleton (or) a 2-element set. Hence we need to consider the following cases. 푛−2 Case1: Let n be even. Then + 2 colours are required in an equitable colouring of 푇 . 2 (3,푛) 푛−2 Let풞 = {푐 , 푐 ,푐 ,……,푐 푐 , 푐 } 푏푒 푎푛 equitable colouring of 푇 , where k = . 1 2 3 푘 푘+1 푘+2 (3,푛) 2 Note that the colour classes of 푐1, 푐2,푐3,……,푐푘 are 2-element sets. While the colour classes of 푐푘+1 and 푐푘+2are singleton sets. Let X be the random variable which represents the colour of an arbitrarily chosen vertex of 푇(3,푛). Then the corresponding p. m. f of G is, 2 푛−2 , 푓표푟 = 1,2, … … . 푛 2 1 푛−2 푛−2 푓() = p(X = i) = { , 푓표푟 = + 1, + 2 푛 2 2 0 , 푒푙푠푒푤ℎ푒푟푒 Hence, the corresponding equitable colouring mean of the tadpole graph 푇(3,푛) is given by, 푛−2 2 푛−2 푛−2 1 푛2+2푛+4 휇 (푇 ) = (1 + 2 + 3 + ⋯ + ) . + ( + 1 + + 2). = 푎푛푑 the 휒푒 (3,푛) 2 푛 2 2 푛 4푛 Volume 10 Issue 1 - 2020 893 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741 Corresponding equitable colouring variance of 푇(3,푛) is, (푛−2)2 2 푛−2 푛−2 1 푛2+2푛+4 휎2 (푇 ) = (12 + 22 + 32 + ⋯ + ) . +[( + 1)2+( + 2)2)]. − ( )2 = 휒푒 (3,푛) 4 푛 2 2 푛 4푛 푛4+20푛2−48 . 48푛2 푛−1 Case 2. Let n be odd. Then, + 1 colours are required in an equitable colouring of 푇 . 2 (3,푛) 푛−1 If {푐 , 푐 ,푐 ,……,푐 , 푐 } be an equitable colouring of 푇 . Where k = . Note that the 1 2 3 푘 푘+1 (3,푛) 2 colour classes of 푐1, 푐2,푐3,……,푐푘 are 2-element sets, while the colour class of 푐푘+1 is a singleton set. Let X be the random variable which represents the colour of an arbitrarily chosen vertex of 푇(3,푛).Then the corresponding p.m.f of G is, 2 푛−1 , 푓표푟 = 1,2, … … . 푛 2 1 푛−1 푓() = p(X = i) = { , 푓표푟 = + 1 푛 2 0, 푒푙푠푒푤ℎ푒푟푒 Then, the corresponding equitable colouring mean is, 푛−1 2 푛−1 1 푛2+2푛+1 휇 (푇 ) = (1 + 2 + 3 + ⋯ + ) . + ( + 1). = and 휒푒 (3,푛) 2 푛 2 푛 4푛 (푛−1)2 2 푛−1 1 푛2+2푛+1 푛4+2푛2−3 휎2 (푇 ) = (12 + 22 + 32 + ⋯ + ) . +( ( + 1)2). − ( )2 = . 휒푒 (3,푛) 4 푛 2 푛 4푛 48푛2 Theorem 2.For a windmill graph 푊푑(4, n) = 퐶푛+퐿1, we have (푛+2)2 , 푓 푛 푠 푒푣푒푛 4(푛+1) 휇 (푊 (4, 푛)) = { 휒푒 푑 푛2+4푛+19 , 푓 푛 푠 표푑푑 4(푛+1) 푛4+4푛3+8푛2+8푛 , 푓 푛 푠 푒푣푒푛 48(푛+1)2 휎2 (푊 (4, 푛)) = { 휒푒 푑 푛4+4푛3+98푛2+188푛−675 , 푓 푛 푠 표푑푑 48(푛+1)2 Proof. Let v be the central vertex of the windmill graph 푊푑(4, n) = 퐶푛+퐿1 and 푣1,푣2, … 푣푛 be the vertices of the cycle 퐶푛. Since v is adjacent to all other vertices of 푊푑, none of them can have the same colour of v. Hence, every other colour class, with respect to an equitable colouring of 푊푑 must be a singleton (or) a 2-element set.
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