(untitled) (Communications in Mathematical Research, 2021)
Let G be a multigraph with vertex set V(G). Assume that a positive integer f(v) with 1 ≤ f(v) ≤ d(v) is associated with each vertex v ∈ V. An edge coloring of G is called an f-edge cover-coloring, if each color appears at each vertex v at least f(v) times. Let χ′_fc(G) be the maximum positive integer k for which an f-edge cover-coloring with k colors of G exists. In this paper, we give a new lower bound of χ′_fc(G), which is sharp.