On f-Edge Cover Coloring of Nearly Bipartite Graphs

Let G(V,E) be a graph, and let f be an integer function on V with 1 f(v) d(v) to each vertex v 2 V. An f-edge cover coloring is an edge coloring C such that each color appears at each vertex v at least f(v) times. The f-edge cover chromatic index of G, denoted by 0 fc (G), is the maximum number of colors needed to f-edge cover color G. It is well known that min v2V {b d(v) µ(v) f(v) c 0 fc (G) f, where µ(v) is the multiplicity of v and f = min{b d(v) f(v) c : v 2 V (G)}. If 0 fc = f, then G is of fc-class 1, otherwise G is of fc-class 2. In this paper, we give some new sucient conditions for a nearly bipartite graph to be of fc-class 1.