[r,s,t]-Coloring of K _n,n

Abstract

Let G=(V(G),E(G)) be a simple graph. Given non-negative integers r,s, and t, an [r,s,t]-coloring of 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. We determine χ _r,s,t (K _n,n ) in all cases.