Distance Graphs andT-Coloring

We discuss relationships amongT-colorings of graphs and chromatic numbers, fractional chromatic numbers, and circular chromatic numbers of distance graphs. We first prove that for any finite integral setTthat contains 0, the asymptoticT-coloring ratioR(T) is equal to the fractional chromatic number of the distance graphG(Z, D), whereD=T−{0}. This fact is then used to study the distance graphs with distance sets of the formDm, k={1, 2, …, m}−{k}. The chromatic numbers and the fractional chromatic numbers ofG(Z, Dm, k) are determined for all values ofmandk. Furthermore, circular chromatic numbers ofG(Z, Dm, k) for some special values ofmandkare obtained.