Circular Chromatic Number and
Mycielski Graphs |
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Authors: | Email author" target="_blank">Genghua?FanEmail author |
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Institution: | (1) Center for Discrete Mathematics, Fuzhou University, Fuzhou, Fujian, 350002, China |
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Abstract: | As a natural generalization of graph coloring, Vince
introduced the star chromatic number of a graph
G and denoted it by
*(G). Later, Zhu called it circular
chromatic number and denoted it by c(G). Let (G)
be the chromatic number of G.
In this paper, it is shown that if the complement of
G is non-hamiltonian, then
c(G)=(G). Denote by M(G)
the Mycielski graph of G.
Recursively define Mm(G)=M(Mm–1(G)). It was conjectured that if
mn–2, then c(Mm(Kn))=(Mm(Kn)). Suppose that
G is a graph on n vertices.
We prove that if
, then
c(M(G))=(M(G)). Let S be the set of vertices of degree
n–1 in
G. It is proved that if
|S| 3, then
c(M(G))=(M(G)), and if |S| 5, then c(M2(G))=(M2(G)), which implies the known results of
Chang, Huang, and Zhu that if n3, c(M(Kn))=(M(Kn)), and if
n5, then
c(M2(Kn))=(M2(Kn)).* Research supported by Grants from National Science
Foundation of China and Chinese Academy of
Sciences. |
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Keywords: | 05C15 |
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