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 mn−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 )).

This is a preview of subscription content, access via your institution.

Author information



Corresponding author

Correspondence to Genghua Fan.

Additional information

* Research supported by Grants from National Science Foundation of China and Chinese Academy of Sciences.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Fan, G. Circular Chromatic Number and Mycielski Graphs. Combinatorica 24, 127–135 (2004).

Download citation

Mathematics Subject Classification (2000):

  • 05C15