Some Results on Edge Cover Coloring of Double Graphs

Let G be a simple graph with vertex set and edge set   V G   E G . An edge coloring C of G is called an edge cover coloring, if each color appears at least once at each vertex   v V G  . The maximum positive integer k such that G has a k edge cover coloring is called the edge cover chromatic number of G and is denoted by . It is known that for any graph G,   c G    1 c G        . If   G c     , then G is called a graph of CI class, otherwise G is called a graph of CII class. It is easy to prove that the problem of deciding whether a given graph is of CI class or CII class is NP-complete. In this paper, we consider the classification on double graph of some graphs and a polynomial time algorithm can be obtained for actually finding such a classification by our proof.