Circular chromatic numbers of distance graphs with distance sets missing multiples (Q1971805)

Given positive integers \(m\), \(k\), \(s\) with \(m>sk\), let \(D_{m,k,s}\) represent the set \[ \{ 1,2,\ldots ,m\}\backslash \{ k,2k,\ldots ,sk\}. \] The distance graph \(G(Z,D_{m,k,s})\) has as vertex set all integers \(Z\) and edges connecting \(i\) and \(j\) whenever \(|i-j|\in D_{m,k,s}\). This paper investigates chromatic numbers and circular chromatic numbers of the distance graphs \(G(Z,D_{m,k,s})\). By establishing bounds for the circular chromatic number \(\chi _{c}(G(Z,D_{m,k,s}))\) of \(G(Z,D_{m,k,s})\), the values of \(\chi (G(Z,D_{m,k,s}))\) are determined for all positive integers \(m\), \(k\), \(s\) and the values of \(\chi _{c}(G(Z,D_{m,k,s}))\) for some positive integers \(m\), \(k\), \(s\).