[r, s, t; f]-COLORING OF GRAPHS
Yong Yu, Guizhen Liu · Journal of the Korean Mathematical Society · 2011
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.