Journal of the
Korean Mathematical Society JKMS

Let $f$ be a function which assigns a positive integer $f(v)$ to each vertex $v\in V(G)$, let $r$, $s$ and $t$ be non-negative integers. An $f$-coloring of $G$ is an edge-coloring of $G$ such that each vertex $v\in V(G)$ has at most $f(v)$ incident edges colored with the same color. The minimum number of colors needed to $f$-color $G$ is called the $f$-chromatic index of $G$ and denoted by $\chi'_f(G)$. An [$r$, $s$, $t$; $f$]-coloring of a graph $G$ is a mapping $c$ from $V(G)\bigcup E(G)$ to the color set $C=\{0,1,\ldots,k-1\}$ such that $|c(v_i)-c(v_j)|\geq r$ for every two adjacent vertices $v_i$ and $v_j$, $|c(e_i)-c(e_j)|\geq s$ and $\alpha(v_i)\leq f(v_i)$ for all $v_i\in V(G)$, $\alpha\in C$ where $\alpha(v_i)$ denotes the number of $\alpha$-edges incident with the vertex $v_i$ and $e_i$, $e_j$ are edges which are incident with $v_i$ but colored with different colors, $|c(e_i)-c(v_j)|\geq t$ for all pairs of incident vertices and edges. The minimum $k$ such that $G$ has an [$r$, $s$, $t$; $f$]-coloring with $k$ colors is defined as the [$r$, $s$, $t$; $f$]-chromatic number and denoted by $\chi_{r, s, t; f}(G)$. In this paper, we present some general bounds for [$r$, $s$, $t$; $f$]-coloring firstly. After that, we obtain some important properties under the restriction $\min\{r, s, t\}=0$ or $\min\{r, s, t\}=1$. Finally, we present some problems for further research.

Keywords: $f$-coloring, [$r$,$s$,$t$]-coloring, [$r$,$s$,$t$,$f$]-coloring, $f$-total coloring, [$r$,$s$,$t$,$f$]-chromatic number

MSC numbers: 05C15