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? ??(G(Z, Dm, k,s)) ? ?m+sk+ 1s+ 1? +1whenm? (s+ 1)k. In this paper, by establishing bounds for the circular chromatic number ?c(G(Z, Dm,k , s)) of G(Z, Dm, k,s), we determine the values of ?(G(Z, Dm,k , s)) for all positive integers m, k, s and?c (G(Z, Dm, k,s)) for some positive integers m, k, s.

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