2-Distance Colorings of Integer Distance Graphs

Abstract

A 2-distance k-coloring of a graph G is a mapping from V (G) to the set of colors {1,. .. , k} such that every two vertices at distance at most 2 receive distinct colors. The 2-distance chromatic number $\chi$ 2 (G) of G is then the mallest k for which G admits a 2-distance k-coloring. For any finite set of positive integers D = {d 1 ,. .. , d k }, the integer distance graph G = G(D) is the infinite graph defined by V (G) = Z and uv $\in$ E(G) if and only if |v -- u| $\in$ D. We study the 2-distance chromatic number of integer distance graphs for several types of sets D. In each case, we provide exact values or upper bounds on this parameter and characterize those graphs G(D) with $\chi$ 2 (G(D)) = {\Delta}(G(D)) + 1.