FROM RAINBOW TO THE LONELY RUNNER: A SURVEY ON COLORING PARAMETERS OF DISTANCE GRAPHS
Daphne Der‐Fen Liu · Taiwanese Journal of Mathematics · 2008
Motivated by the plane coloring problem, Eggleton, Erd\H{o}s and Skelton initiated the study of distance graphs. Let $D$ be a set of positive integers. The distance graph generated by $D$, denoted by $G(\mathbb{Z}, D)$, has all integers $\mathbb{Z}$ as the vertex set, and two vertices $x$ and $y$ are adjacent whenever $|x-y| \in D$. The chromatic number, circular chromatic number and fractional chromatic number of distance graphs have been studied extensively in the past two decades; these coloring parameters are also closely related to some problems studied in number theory and geometry. We survey some research advances and open problems on coloring parameters of distance graphs.