On f-Edge Cover Coloring of Regular Graphs

Let G be a graph, and let f be an integer function on V with 1 <= f (v) <= d(v) to each vertex v is an element of V. An f-edge cover coloring is a coloring of edges of E(G) such that each color appears at each vertex V is an element of V(G) at least f(v) times. The maximum number of colors needed to f-edge cover color G is called the f-edge cover chromatic index of G and denoted by X'(fc)(G). It is well known that any simple graph G has the f-edge cover chromatic index equal to delta(f)(G) or delta(f)(G) - 1, where delta(f)(G) = min{left perpendiculard(v)/f(v)right perpendicular : v is an element of V(G)}. If X'(fc) = delta(f) (G), then G is of f(c)-class 1, otherwise G is of f(c)-class 2. In this paper two sufficient conditions for a regular graph to be of f(c)-class 1 or f(c)-class 2 are obtained and two necessary and sufficient conditions for a regular graph to be of f(c)-class 1 are also presented.