Coloring of distance graphs with intervals as distance sets

Abstract

Let $D$ be a set of positive integers. The distance graph $G(Z,D)$ with distance set $D$ is the graph with vertex set $Z$ in which two vertices $x,y$ are adjacent if and only if $|x-y|\in D$. The fractional chromatic number, the chromatic number, and the circular chromatic number of $G(Z,D)$ for various $D$ have been extensively studied recently. In this paper, we investigate the fractional chromatic number, the chromatic number, and the circular chromatic number of the distance graphs with the distance sets of the form $D_{m,[k,k^{\prime }]}=\{1,2,\dots ,m\}-\{k,k+1,\dots ,k^{\prime }\}$, where $m$, $k$, and $k^{\prime }$ are natural numbers with $m\ge k^{\prime }\ge k$. In particular, we completely determine the chromatic number of $G(Z,D_{m,[2,k^{\prime }]})$ for arbitrary $m$, and $k^{\prime }$.