On f-edge cover-coloring of simple graphs

Let G(V, E) be a simple graph, and let f be an integer function on V with 1 ≤ f(v) ≤ d(v) to each vertex v ε V. An f-edge cover-coloring of a graph G is a coloring of edge set E such that each color appears at each vertex v ε 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 such that an f-edge cover-coloring of G exists. Any simple graph G has f-edge cover chromatic index equal to δ_f or δ_f-1, where δ_f = min vεV{[d(v/f(v]}. If χ′_fc(G) = δ_f, then G is of C_f I class; otherwise G is of C_f II class. In this paper, we give some sufficient conditions for a graph to be of C_f I class, and discuss the classification problem of complete graphs on f-edge cover-coloring.