Asymptotic Clique Covering Ratios of Distance Graphs

Abstract

Given a finite set D of positive integers, the distance graph G(Z , D) has Z as the vertex set and {i j : |i − j| ∈ D} as the edge set. Given D, the asymptotic clique covering ratio is defined as (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.