Circular chromatic number and Mycielski graphs (Q5920576)

Some results of \textit{G. Fan} [Combinatorica 24, No. 1, 127--135 (2004; Zbl 1056.05057)] regarding the circular chromatic numbers of Mycielski graphs are improved. For instance, Fan showed that if an \(n\)-vertex graph \(G\) has at least three vertices of degree \(n-1\) then \(\chi _{c}(M(G))=\chi (M(G))\). The present author observes that if \(G\) contains a 3-clique (resp. 4-clique) whose vertices all have degree \(n-2\) (resp. \(\geq n-3\)), and \(G\) has at least one vertex not adjacent to any vertex of the clique, then \(\chi _{c}(M(G))=\chi (M(G))\). Fan proved that if \(G\) has at least five vertices of degree \(n-1\) then \(\chi _{c}(M^{2}(G))=\chi (M^{2}(G))\), and the present author shows that if \(G\) has four vertices of degree \(n-1\) then \(\chi _{c}(M^{2}(G))=\chi (M^{2}(G))\).