2-Distance Colorings of Integer Distance Graphs

TL;DR

This paper investigates the 2-distance coloring of integer distance graphs, providing exact values and bounds for their chromatic numbers, and characterizing graphs with maximum possible 2-distance chromatic number.

Contribution

It offers new results on the 2-distance chromatic number for various integer distance graphs, including exact values, bounds, and characterizations.

Findings

Exact values and bounds for 2-distance chromatic numbers.

Characterization of graphs with maximum 2-distance chromatic number.

Analysis for specific sets D of positive integers.

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.