Circular Chromatic Numbers of Distance Graphs with Distance Sets Missing Multiples

Given a set D of positive integers, the distance graph G(Z , D) has all integers as vertices, and two vertices are adjacent if and only if their difference is in D; that is, the vertex set is Z and the edge set is {uv : |u − v| ∈ D}. We call D the distance set. This paper studies chromatic and circular chromatic numbers of some distance graphs with certain distance sets. The circular chromatic number of a graph is a natural generalization of the chromatic number of a graph, introduced by Vince [15] as the name “star chromatic number.” Suppose p and q are positive integers such that p ≥ 2q . Let G be a graph with at least one edge. A (p, q)-coloring of G = (V, E) is a mapping c from V to {0, 1, . . . , p − 1} such that q ≤ |c(x)− c(y)| ≤ p− q for any edge xy in E . The circular chromatic number χc(G) of G is the infimum of the ratios p/q for which there exists a (p, q)-coloring of G. Note that for p ≥ 2, a (p, 1)-coloring of a graph G is simply an ordinary p-coloring of G. Therefore, χc(G) ≤ χ(G) for any graph G. Let G be a graph which is not a null graph. On the other hand, it has been shown [15] that for all finite graphs G, we have χ(G) − 1 < χc(G). Applying a result of de Bruijn and Erdős [6], this can be proved also for infinite graphs. Therefore, χ(G) = dχc(G)e if G 6= Nn . In particular, two graphs with the same circular chromatic number also have the same chromatic number. However, two graphs with the same chromatic number may have different circular chromatic numbers. Thus χc(G) is a refinement of χ(G), and it contains more information about the structure of the graph. It is usually much more difficult to determine the circular chromatic number of a graph than to determine its chromatic number. The fractional chromatic number of a graph is another well-known variation of the chromatic number. A fractional coloring of a graph G is a mapping c from I(G), the set of all independent sets of G, to the interval [0, 1] such that ∑