Circular Chromatic Numbers of Distance Graphs with Distance Sets Missing Multiples

Given positive integers m, k, s withm & sk, let Dm, k, srepresent the set {1, 2,⋯ , m } \ { k, 2 k,⋯ , sk }. The distance graph G(Z, Dm, k, s) has as vertex set all integersZ and edges connecting i and j whenever | i − j | ∈ Dm, k, s. This paper investigates chromatic numbers and circular chromatic numbers of the distance graphs G(Z, Dm, k,s). Deuber and Zhu and Liu have shown that⌈m + sk + 1s + 1⌉ ≤ χ(G(Z, Dm, k,s)) ≤ ⌈m + sk + 1s + 1⌉ + 1whenm ≥ (s + 1)k. In this paper, by establishing bounds for the circular chromatic number χc(G(Z, Dm,k , s)) of G(Z, Dm, k,s), we determine the values of χ(G(Z, Dm,k , s)) for all positive integers m, k, s andχc (G(Z, Dm, k,s)) for some positive integers m, k, s.