Circular chromatic numbers of a class of distance graphs

AbstractSuppose m,k,s are positive integers with m>sk. Let Dm,k,s denote the set {1,2,…,m}⧹{k,2k,…,sk}. The distance graph G(Z,Dm,k,s) has as vertex set all integers Z and edges connecting i and j whenever |i−j|∈Dm,k,s. This paper determines the circular chromatic number of all the distance graphs G(Z,Dm,k,s)

Planar Graphs with Circular Chromatic Numbers between 3 and 4

AbstractThis paper proves that for every rational number r between 3 and 4, there exists a planar graph G whose circular chromatic number is equal to r. Combining this result with a recent result of Moser, we arrive at the conclusion that every rational number r between 2 and 4 is the circular chromatic number of a planar graph