Some properties on f-edge covered critical 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 ∈ 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 ∈ 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 an f-edge cover chromatic index equal to δf or δf - 1, where δf = minv∈V{⌊d(v)/f(v)⌋}. Let G be a connected and not complete graph with χ′fc(G) = δf - 1, if for each u, v ∈ V and e = uv ∉ E, we have χ′fc(G + e) > χ′fc(G), then G is called an f-edge covered critical graph. In this paper, some properties on f-edge covered critical graph are discussed. It is proved that if G is an f-edge covered critical graph, then for each u, v ∈ V and e = uv ∉ E there exists w ∈ {u, v} with d(w) ≤ δf(f(w) +1) - 2 such that w is adjacent to at least d(w) - δf + 1 vertices which are all δf - vertex in G. © 2007 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.