Some class 1 graphs on g c -colorings

An edge-coloring of a graph G is an assignment of colors to all the edges of G. A g c -coloring of a graph G is an edge-coloring of G such that each color appears at each vertex at least g(v) times. The maximum integer k such that G has a g c -coloring with k colors is called the g c -chromatic index of G and denoted by ..gc (G). In this paper, we extend a result on edge-covering coloring of Zhang and Liu in 2011, and give a new sufficient condition for a simple graph G to satisfy ..gc (G) = . g (G), where .g(G)=minv.V(G){.d(v)g(v).}.