On $f$-Edge Cover Chromatic Index of Multigraphs

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.