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Differential Coloring of Graphs 163 CORE Metadata, citation and similar papers at core.ac.uk SUT Journal of Mathematics Vol. 47, No. 2 (2011), 161{174 Di®erential Coloring of Graphs P. Roushini Leely Pushpam, I. Sahul Hamid and D. Yokesh (Received March 11, 2011; Revised September 18, 2011) Abstract. Let G = (V; E) be a connected graph. For a set S ⊆ V , the di®erential of S, denoted by @(S), is de¯ned to be @(S) = jB(S)j ¡ jSj, where B(S) = N(S) \ (V ¡ S). A set S ⊆ V is said to be a positive di®erential set if @(S) ¸ 0. A partition fV1;V2;:::;Vkg of V is said to be a positive di®erential chromatic partition of G if each Vi is both independent and positive di®erential. The minimum order of a positive di®erential chromatic partition of G is called the di®erential chromatic number of G and is denoted by Â@ (G). In this paper we initiate a study of this parameter. AMS 2010 Mathematics Subject Classi¯cation. 05C. Key words and phrases. Di®erential coloring, di®erential chromatic number. x1. Introduction By a graph G = (V; E), we mean a ¯nite, undirected graph with neither loops nor multiple edges. For graph theoretic terminology we refer to Chartrand and Lesniak [1]. All graphs in this paper are assumed to be connected with at least two vertices. For a vertex v 2 V , the open neighborhood of v is the set N(v) = fu 2 V juv 2 Eg and the closed neighborhood is the set N[v] = N(v) [ fvg. For a set S ⊆ V , its open neighborhood is N(S) = [v2SN(u) and the closed neighborhood is N[S] = N(S) [ S. The subgraph induced by S is denoted as G[S]. Consider the game where you are allowed to buy as many tokens as you like at a cost of $1 each. For example, suppose that you buy k tokens. You then place the tokens on some subset of k vertices of G. For each vertex of G which has no token on it, but is adjacent to a vertex with a token on it, you receive $1 from the bank. Your objective is to maximize your pro¯t, that is, the total value received from the bank minus the cost of the tokens bought. Notice that you do not receive any credit for the vertices in which you place a token. 161 162 P. R. L. PUSHPAM, I. S. HAMID AND D. YOKESH The above game can be mathematically formulated as follows. For a set S ⊆ V , the boundary B(S) of S is to be de¯ned B(S) = N(S) \ (V ¡ S) and then the di®erential @(S) of S is to be de¯ned @(S) = jB(S)j ¡ jSj. Now, the di®erential of the graph G, denoted by @(G), is de¯ned to be @(G) = maxf@(S): S ⊆ V g. Thus, for the given graph G, maximizing the pro¯t as mentioned in the game is equivalent to determining the value of @(G). The di®erential @(S) of a set was introduced by Hedetniemi [4] and also was considered by Goddard and Henning [3] who denoted it by ´(S). Obviously, the di®erential of a set in a graph may be positive or negative or may be zero. However, since it has been proved in [5] that @(G) ¸ ¢(G) ¡ 1, it follows that the di®erential of a graph without isolated vertices is either positive or zero. Hence one need not consider the sets of negative di®erential while determining @(G) and so the sets of zero or positive di®erential play a vital role in the game. Motivated by this observation we take interest on such sets, that is, the sets of zero or positive di®erential and call these sets as positive di®erential set. (Here we use the term \positive di®erential set" in the sense that by the choice of those sets where the tokens bought are to be placed, one will not lose money, whether or not one gains money). However, our main focus in this paper is not on the study of the positive di®erential sets, but on the problem of partitioning V into independent positive di®erential sets, that is, positive di®erential sets in which no two vertices are adjacent. That is, we consider the partition of V into independent positive di®erential sets and we call this partition as positive di®erential chromatic partition and the minimum order of a positive di®erential chromatic partition of G is called the di®erential chromatic number of G and we denote it by Â@(G). Also we use the term Â@-partition to denote a positive di®erential partition of order Â@. In this paper we initiate a study of this parameter. x2. Di®erential Chromatic Number In this section, we determine the value of the di®erential chromatic number for some families of graphs such as paths, cycles, complete graphs, wheels, complete multipartite graphs and complete binary trees. Since it is straightforward to determine the value of the di®erential chro- matic number for the above mentioned graphs, we just state them without proof. Proposition 1. (i) For the paths Pn and cycles Cn on n vertices, ( 3 if n is odd; Â@(Pn) = Â@(Cn) = 2 if n is even: DIFFERENTIAL COLORING OF GRAPHS 163 (ii) If Wn is a wheel on n vertices, then ( 4 if n is even; Â@(Wn) = 3 if n is odd: Proposition 2. Let G = Km1;m2;:::;mk be a complete k-partite graph with Xk m1 ¸ m2 ¸ ¢ ¢ ¢ ¸ mk ,mk ¸ 1,k ¸ 2 and let N = mi. Then i=2 8 » ¼ < m ¡ N k + 1 if N < m ; N 1 Â@(G) = : k otherwise. Proof. Let G be a complete k-partite graph with k-partition (X1;X2;:::;Xk). Let jXij = mi, where i = 1; 2; : : : ; k. Clearly each Xi, where 2 · i · k, is a positive di®erential set. Now, if N ¸ m1, then X1 is also a positive di®erential set so that the k-partition (X1;X2;:::;Xk) of G itself is a minimum positive di®erential chromatic partition of G. Hence Â@(G) = k. If N < m1, f g ff g f g let X1 = x1; x2; : : : ; xm1 . Then x1; x2; : : : ; xN , xN+1; xN+2; : : : ; x2N , fx2N+1; x2N+2; : : : ; x3N g, fx3N+1; x3N+2§ ; : : : ; x¨4N g, ::: , fxrN ; xrN+1, xrN+2, ¡ g g m1 N ::: , xm1 ;X2;X3;:::;Xk , where r = N is clearly a» minimum¼ positive m ¡ N di®erential chromatic partition of G. Hence  (G) = k + 1 . @ N Corollary 1. For a complete graph Kn on n vertices, Â@(Kn) = n. Proposition 3. If G is a complete binary tree, then Â@(G) = 3. Proof. Let G be a k-level complete binary tree, where k ¸ 1. Let vij denote the jth vertex in the ith level. We consider the following cases. Case 1. k is odd k k Let A = fvkj : j = 1; 2; 5; 6; 9; 10;:::; 2 ¡ 3; 2 ¡ 2g; k B = fvk¡1;j : j = 2; 4; 6;:::; 2 ¡ 1g; m C = fvmj : 0 · m · k and 1 · j · 2 ¡ 3; where m is oddg; V1 = A [ B [ C; m V2 = fvmj : 0 · m · k and 1 · j · 2 ¡ 2; where m is eveng; V3 = V ¡ fV1 [ V2g: 164 P. R. L. PUSHPAM, I. S. HAMID AND D. YOKESH Case 2. k is even k k Let A = fvkj : j = 1; 2; 5; 6; 9; 10;:::; 2 ¡ 4; 2 ¡ 3g; k B = fvk¡1;j : j = 2; 4; 6;:::; 2 ¡ 1g; V1 = A [ B; m C = fvmj : 0 · m · k and 0 · j · 2 ¡ 3; where m is oddg; D = fvk;k¡2g; V2 = C [ D; V3 = V ¡ fV1 [ V2g: Thus in both the cases fV1;V2;V3g forms a positive di®erential chromatic partition of G and hence Â@(G) · 3. Now, since a complete binary tree is a bipartite graph with parts of unequal size, we have Â@(G) ¸ 3 (see Theorem 1). Thus Â@(G) = 3. x3. Bounds on Â@(G) In this section, we establish some bounds on Â@(G) in terms of order of the graph and some well known parameters such as chromatic number, indepen- dent domination number, induced paired domination number and packing number. Further we obtain an upper bound for trees in terms of the maxi- mum degree. All graphs considered in this section are of order n. Theorem 1. For any graph G, we have 2 · Â@(G) · n. Further, Â@(G) = 2 if and only if G is a bipartite graph with parts of equal size and Â@(G) = n if and only if G is either a star or a complete graph. Proof. Obviously 2 · Â@(G) · n. Suppose Â@(G) = 2. Let fV1;V2g be a positive di®erential partition of G. Now, since V1 is a positive di®erential set, we have jN(V1)j ¸ jV1j and since jN(V1)j · jV2j, it follows that jV2j ¸ jV1j. Similarly jV1j ¸ jV2j and hence jV1j = jV2j. Also if G is a bipartite graph with parts of equal size, then Â@(G) = 2. Now, suppose Â@(G) = n. Then G has no independent positive di®erential set with more than one vertex. If G is a star we are through. Otherwise we claim that G is complete. Suppose G has two non-adjacent vertices, say u and v. If x and y are neighbours of u and v respectively, then x = y, for otherwise fu; vg will be an independent positive di®erential set in G.
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