From rainbow to the lonely runner: A survey on coloring parameters of distance graphs

Motivated by the plane coloring problem, Eggleton, Erdos 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(Z, D), has all integers Z as the vertex set, and two vertices x and y are adjacent whenever vertical bar x - y vertical bar is an element of 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.