Xia Zhang

An f-coloring of a graph G is a coloring of edges of $E(G)$ such that each color appears at each vertex $v\in V(G)$ at most $f(v)$ times. The minimum number of colors needed to $f$-color $G$ is called the $f$-chromatic index $\chi'_{f}(G)$ of $G$. Any simple graph $G$ has $f$-chromatic index equal to $\Delta_{f}(G)$ or $\Delta_{f}(G)+1$, where $\Delta_{f}(G)=\max\_limits^{}_{v\in V}\{\lceil\frac{d(v)}{f(v)}\rceil\}$. If $\chi'_{f}(G)=\Delta_{f}(G)$, then $G$ is of $C_{f}$ 1; otherwise $G$ is of $C_{f}$ 2. Some conditions are given based on the $f$-core of $G$ (i.e., the subgraph of $G$ induced by the vertices of $V^{*}_{0}=\{v:\Delta_f(G)=\frac{d(v)}{f(v)},v\in V\}$), which are sufficient for $G$ to be of $C_{f}$ 1. Also, the $f$-chromatic indices of some special classes of simple graphs are obtained. By virtue of the relation between $f$-coloring and equitable edge-coloring, we obtain a sufficient condition for equitable edge-colorings of simple graphs, which substantially extends the previous result of Hilton and de Werra (Discrete Math., 128(1994), 179--201).