On f-edge cover coloring of regular graphs - Li, Jinbo; Liu, Guizhen (Utilitas Mathematica, 2011)
Let G be a graph, and let f be an integer function on V with 1 ≤ f(v) ≤ d(v) to each vertex v G V. An f-edge cover coloring is a coloring of edges of E[G) such that each color appears at each vertex v ∈ 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 χ′fc(G). It is well known that any simple graph G has the f-edge cover chromatic index equal to δf(G) or δf(G) - 1, where δf(G) = min{[d(u)/f(v)] : v ∈ V(G)). If χ′fc = δf(G), then G is of f c-class 1, otherwise G is of fc-class 2. In this paper two sufficient conditions for a regular graph to be of fc-class 1 or fc-class 2 are obtained and two necessary and sufficient conditions for a regular graph to be of fc-class 1 are also presented.