Asymptotic Clique Covering Ratios of Distance Graphs

Given a finite set D of positive integers, the distance graph G(Z, D) has Z as the vertex set and { ij: | i − j | ∈ D } as the edge set. Given D, the asymptotic clique covering ratio is defined as S(D) = lim supn → ∞ncl(n), where cl(n) is the minimum number of cliques covering any consecutive n vertices of G(Z, D). The parameter S(D) is closely related to the ratio spT(G)χ(G)of a graph G, where χ(G) and spT(G) denote, respectively, the chromatic number and the optimal span of a T -coloring of G. We prove that for any finite set D, S(D) is a rational number and can be realized by a ‘periodical’ clique covering of G(Z, D). Then we investigate the problem for which sets D the equality S(D) = ω(G(Z, D)) holds. (In general, S(D) ≤ ω(G(Z, D)), where ω(G) is the clique number of G.) This problem turns out to be related to T -colorings and to fractional chromatic number and circular chromatic number of distance graphs. Through such connections, we shall show that the equality S(D) = ω(G(Z, D)) holds for many classes of distance graphs. Moreover, we raise questions regarding other such connections.