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 χ _c (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 χ _c (G)=χ(G). Denote by M(G) the Mycielski graph of G. Recursively define M ^m(G)=M(M ^m−1(G)). It was conjectured that if m≤n−2, then χ _c (M ^m(K _n ))=χ(M ^m(K _n )). Suppose that G is a graph on n vertices. We prove that if $\chi {\left( G \right)} \geqslant \frac{{n + 3}} {2}$ , then χ _c (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 χ _c (M(G))=χ(M(G)), and if |S|≥ 5, then χ _c (M ²(G))=χ(M ²(G)), which implies the known results of Chang, Huang, and Zhu that if n≥3, χ _c (M(K _n ))=χ(M(K _n )), and if n≥5, then χ _c (M ²(K _n ))=χ(M ²(K _n )).