Circular Chromatic Numbers and Fractional Chromatic Numbers of Distance Graphs with Distance Sets Missing An Interval

Suppose S is a subset of a metric spaceM with a metric δ, and D a subset of positive real numbers. The distance graph G(S, D), with a distance set D, is the graph with vertex set S in which two vertices x and y are adjacent iff δ(x, y) ∈ D. Distance graphs, first studied by Eggleton et al. [7], were motivated by the well-known plane-coloring problem: What is the minimum number of colors needed to color all points of a euclidean plane so that points at unit distances are colored with different colors. This problem is equivalent to determining the chromatic number of the distance graph G(R2, {1}). It is well-known that the chromatic number of this distance graph is between 4 and 7 (see [12, 15]). However, the exact number of colors needed remains unknown. For distance graphs on the real line R or the integer set Z , the problem of finding the chromatic numbers of G(R, D) or G(Z , D) for different D sets has been studied extensively (see [3, 10, 13, 14, 17, 18, 20, 22]). Two recent papers [3, 14] related distance graphs to the T coloring problem. Chromatic numbers and fractional chromatic numbers of distance graphs were used to derive bounds for T -spans of the corresponding T -colorings, and vice versa. In this paper, we study circular chromatic numbers and fractional chromatic numbers of distance graphs G(Z , D) for various D sets. The circular chromatic number of a graph is a natural generalization of the chromatic number of a graph, introduced by Vince [16] under the name the ‘star chromatic number’ of a graph. Suppose p and q are positive integers such that p ≥ 2q. A (p, q)-coloring of a graph G = (V, E) is a mapping c from V to {0, 1, . . . , p − 1} such that ‖c(x) − c(y)‖p ≥ q for any edge xy in E , where ‖a‖p = min{a, p− a}. The circular chromatic number χc(G) of G is the infimum of the ratios p/q for which there exist (p, q)-colorings of G. Note that 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. On the other hand, it has been shown [16] that for all graphs G, we have χ(G)− 1 < χc(G). Therefore, χ(G) = dχc(G)e. 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 main results of this article determine the circular chromatic numbers of various distance graphs. These results may be viewed as improvements