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 )).

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Correspondence to Genghua Fan.

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

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Fan, G. Circular Chromatic Number and Mycielski Graphs. Combinatorica 24, 127–135 (2004). https://doi.org/10.1007/s00493-004-0008-9

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Mathematics Subject Classification (2000):

  • 05C15