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 (in) 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 X´fc (G), is the maximum number of colors needed to f-edge cover color G. It is well-known that min(v (in) V) [d(v) - μ(v)/ f(v)] ≤ X´fc (G) ≤ δf, where μ(v) is the multiplicity of v and δf= min{d(v)/ f(v)⌡ : v (in) V (G)}. If X´fc (G) = δf, then G is of fc-class 1, otherwise G is of fc-class 2. In this paper, we give some new su cient conditions for a nearly bipartite graph to be of fc-class 1.