[r, s, t; f]-COLORING OF GRAPHS

Abstract

Let f be a function which assigns a positive integer f(v)to each vertex v 2 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 2 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 <수식>-coloring of a graph G is a mapping c from <수식> to the color set C = {0, 1,...,k-1}such that <수식> for every two adjacent vertices vi and vj ,<수식> for all vi 2 V (G), <수식> where α(vi) denotes the number of α-edges incident with the vertex vi and ei,ej are edges which are incident with vi but colored with different colors,<수식> 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 Χ_(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.