A generalization of chromatic index

Let G = (V, E) be a graph and k > 2 an integer. The general chromatic index x;(G) of G is the minimum order of a partition P of E such that for any set F in P every component in the subgraph (F) induced by F has size at most k1. This paper initiates a study of x;(G) and generalizes some known results on chromatic index. The purpose of this paper is to obtain a generalization of chromatic index. Compared to many generalizations of chromatic number, there exist very few generalizations of chromatic index in the literature. For example, see [2] and [3]. Let G=( V, E) be a graph and k 22 an integer. A set FcE is and Zk-set (or k-independent set) if every component in the subgraph (F) induced by F has size at most k 1. Equivalently, a set F c E is k-independent if the sum of the degrees of the vertices in every component of the subgraph (F) is r, where 2 d r g2 (k 1). A partition {E 1, E,, . . . , E,] of E is an I,-partition if each Ei is an &-set. An I,-edge coloring of G is a coloring of the edges of G so that the set of all edges receiving the same color is an Zk-set. An I,-edge coloring which uses r colors is called a (k,r)-edge coloring. The k-chromatic index xi = x;(G) of G is the minimum number of colors needed in an I,-edge coloring of G. If x;(G) = n, then G is said to be (k, n)-edge chromatic. The k-edge independence number Plk=Plk(G) of G is the maximum cardinality of an I,-set. Clearly, if M is any independent set of edges, then M is an Ik-set for all k 3 2. We observe that x;(G)=x’(G), the chromatic index. Also PI2 =/?r, the edge independence number of G. If G has size q, then x;(G)= 1 for all k>q. If L(G) is the line graph of G, then x’(G) = XV(G)) where x(L(G)) is the chromatic number of L(G). Correspondence to: E. Sampathkumar, Department of Mathematics, Mysore University, Mysore 570006, India. 0012-365X/94/$07.00