The Classification of Regular Graphs on $f$-colorings

Abstract

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 of $G$ and denoted by ${\chi }_f^{\prime }(G)$. Any simple graph $G$ has the $f$-chromatic index equal to ${\mathrm{\Delta }}_f(G)$ or ${\mathrm{\Delta }}_f(G)+1$, where ${\mathrm{\Delta }}_f(G)=\underset{v\in V}{max}\{\lceil \frac{d(v)}{f(v)}\rceil \}$. If ${\chi }_f^{\prime }(G)={\mathrm{\Delta }}_f(G)$, then $G$ is of $C_f$ $1$; otherwise $G$ is of $C_f$ $2$. In this paper, two sufficient conditions for a regular graph to be of $C_f$ $1$ or $C_f$ $2$ are obtained and two necessary and sufficient conditions for a regular graph to be of $C_f$ $1$ are also presented.