Distance graphs with missing multiples in the distance sets

Given positive integers m, k and s with m > ks, let Dm,k,s represent the set {1, 2, · · · ,m} − {k, 2k, · · · , sk}. The distance graph G(Z,Dm,k,s) has as vertex set all integers Z and edges connecting i and j whenever |i − j| ∈ Dm,k,s. The chromatic number and the fractional chromatic number of G(Z,Dm,k,s) are denoted by χ(Z,Dm,k,s) and χf (Z,Dm,k,s), respectively. For s = 1, χ(Z,Dm,k,1) was studied by Eggleton, Erdős and Skilton [6], Kemnitz and Kolberg [12], and Liu [13], and was solved lately by Chang, Liu and Zhu [2] who also determined χf (Z,Dm,k,1) for any m and k. This article extends the study of χ(Z,Dm,k,s) and χf (Z,Dm,k,s) to general values of s. We prove χf (Z,Dm,k,s) = χ(Z,Dm,k,s) = k if m < (s + 1)k; and χf (Z,Dm,k,s) = (m + sk + 1)/(s + 1) otherwise. The latter result provides a good lower bound for χ(Z,Dm,k,s). A general upper bound for χ(Z,Dm,k,s) is found. We prove the upper bound can be improved to d(m + sk + 1)/(s + 1)e + 1 for some values of m, k and s. In particular, when s + 1 is prime, χ(Z,Dm,k,s) is either d(m + sk + 1)/(s + 1)e or d(m + sk + 1)/(s + 1)e + 1. By using a special coloring method called the pre-coloring method, many distance graphs G(Z,Dm,k,s) are classified into Supported in part by the National Science Foundation under grant DMS 9805945. Supported in part by the National Science Council under grant NSC87-2115-M110-004.