Generalized Mycielski’s graphs (also known as cones over graphs) are the natural generalization of Mycielski’s graphs (which were first introduced by Mycielski  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 . 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.
|Number of pages||10|
|Journal||Journal of Nanjing University (Mathematical Biquarterly)|
|Publication status||Published - Nov 2006|