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 ∈ 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 χ ′ fc(G), is the maximum number of colors needed to f -edge cover color G. It is well known that min v∈V {bd(v)− μ(v) f(v) c ≤ χ′fc(G) ≤ δf , where μ(v) is the multiplicity of v and δf = min{b d(v) f(v)c : v ∈ V (G)}. If χ ′ fc = δf , then G is of fc-class 1, otherwise G is of fc-class 2. In this paper, we give some new sufficient conditions for a nearly bipartite graph to be of fc-class 1.