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 sup n→∞ n cl(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. Supported in part by the National Science Foundation under grant DMS 9805945. Supported in part by the National Science Council, R. O. C., under grant NSC87-2115-M110004.