Circular Chromatic Number and Mycielski Graphs

As a natural generalization of graph coloring, Vince introduced the star chromatic number of a graph G and denoted it by χ∗(G). Later, Zhu called it circular chromatic number and denoted it by χc(G). Let χ(G) be the chromatic number of G. In this paper, it is shown that if the complement of G is non-hamiltonian, then χc(G)=χ(G). Denote by M(G) the Mycielski graph of G. Recursively define Mm(G)=M(Mm−1(G)). It was conjectured that if m≤n−2, then χc(M(Kn))=χ(M(Kn)). Suppose that G is a graph on n vertices. We prove that if χ(G)≥ n+3 2 , then χc(M(G))=χ(M(G)). Let S be the set of vertices of degree n−1 in G. It is proved that if |S| ≥ 3, then χc(M(G))=χ(M(G)), and if |S| ≥ 5, then χc(M (G))=χ(M(G)), which implies the known results of Chang, Huang, and Zhu that if n≥3, χc(M(Kn))=χ(M(Kn)), and if n≥5, then χc(M(Kn))=χ(M(Kn)).