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(upsilon) <= d(v) to each vertex upsilon is an element of V. An f-edge cover coloring is an edge coloring C such that each color appears at each vertex v at least f(upsilon) times. The f-edge cover chromatic index of G, denoted by chi(fc)'(G), is the maximum number of colors needed to f-edge cover color G. It is well-known that min upsilon is an element of V left perpendicular d(upsilon)- mu(upsilon)/f(upsilon) right perpendicular <= chi(fc)'(G) <= delta(f), where mu(upsilon) is the multiplicity of upsilon and delta(f) = min{left perpendicular d(upsilon)/f(upsilon) right perpendicular : upsilon is an element of V(G)}. If chi(fc)' = delta(f), then G is of f(c)-class 1, otherwise G is of f(c)-class 2. In this paper, we give some new sufficient conditions for a nearly bipartite graph to be of f(c)-class 1.