Formula omitted]-Colorings of graphs

Abstract

Given non-negative integers r, s, and t, an [ r , s , t ] - coloring of a graph G = ( V ( G ) , E ( G ) ) is a mapping c from V ( G ) ∪ E ( G ) to the color set { 0 , 1 , … , k - 1 } such that | c ( v i ) - c ( v j ) | ⩾ r for every two adjacent vertices v i , v j , | c ( e i ) - c ( e j ) | ⩾ s for every two adjacent edges e i , e j , and | c ( v i ) - c ( e j ) | ⩾ t for all pairs of incident vertices and edges, respectively. The [ r , s , t ] - chromatic number χ r , s , t ( G ) of G is defined to be the minimum k such that G admits an [ r , s , t ] -coloring. This is an obvious generalization of all classical graph colorings since c is a vertex coloring if r = 1 , s = t = 0 , an edge coloring if s = 1 , r = t = 0 , and a total coloring if r = s = t = 1 , respectively. We present first results on χ r , s , t ( G ) such as general bounds and also exact values, for example if min { r , s , t } = 0 or if G is a complete graph.