R, s, t; f]-COLORING OF GRAPHS

Abstract

Let f be a function which assigns a positive integer f(v) to each vertex v <TEX>$\in$</TEX> 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 <TEX>$\in$</TEX> 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 <TEX>${\chi}'_f$</TEX>(G). An [r, s, t; f]-coloring of a graph G is a mapping c from V(G) <TEX>$\bigcup$</TEX> E(G) to the color set C = {0, 1, <TEX>$\ldots$</TEX>; k - 1} such that |c(<TEX>$v_i$</TEX>) - c(<TEX>$v_j$</TEX> )| <TEX>$\geq$</TEX> r for every two adjacent vertices <TEX>$v_i$</TEX> and <TEX>$v_j$</TEX>, |c(<TEX>$e_i$</TEX> - c(<TEX>$e_j$</TEX>)| <TEX>$\geq$</TEX> s and <TEX>${\alpha}(v_i)$</TEX> <TEX>$\leq$</TEX> f(<TEX>$v_i$</TEX>) for all <TEX>$v_i$</TEX> <TEX>$\in$</TEX> V (G), <TEX>${\alpha}$</TEX> <TEX>$\in$</TEX> C where <TEX>${\alpha}(v_i)$</TEX> denotes the number of <TEX>${\alpha}$</TEX>-edges incident with the vertex <TEX>$v_i$</TEX> and <TEX>$e_i$</TEX>, <TEX>$e_j$</TEX> are edges which are incident with <TEX>$v_i$</TEX> but colored with different colors, |c(<TEX>$e_i$</TEX>)-c(<TEX>$v_j$</TEX>)| <TEX>$\geq$</TEX> 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 <TEX>${\chi}_{r,s,t;f}$</TEX> (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.