Circular chromatic numbers of the generalized Mycielskians of cycles

Generalized Mycielski’s graphs (also known as cones over graphs) are the natural generalization of Mycielski’s graphs (which were first introduced by Mycielski [16] in 1955). Given a graph G and any integer m≥0, one can transform G into a new graphμm(G), the generalized Mycielskian of G. Many basic properties ofμm(G) were established in [14,19]. And the circular chromatic numbers of the generalized Myciel-skians of complete graphs were completely determined in [12]. Here we determine the circular chromatic numbers ofμm(Cn) for any m≥0 and n≥3, where Cn is the cycle of length n. We also investigate the circular chromatic numbers ofμm(Cn) - v for each vertex v ofμm(G). For m≥1 and odd n(≥5), the graphsμm(Cn) turn out to be 4-critical, triangle-free graphs with Xc = X = 4. Furthermore the odd girth of graphsμm(Cn) can be arbitrarily large.