The circular chromatic number of the Mycielskian of G

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), we now call the Mycielskian of G, which has the same clique number as G and whose chromatic number equals χ(G)+1. Chang, Huang, and Zhu [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear] have investigated circular chromatic numbers of Mycielskians for several classes of graphs. In this article, we study circular chromatic numbers of Mycielskians for another class of graphs Gk. The main result is that χc(μ(G d k)) = χ(μ(G d k)), which settles a problem raised in [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear, and X. Zhu, to appear]. As χc(Gk) = k d and χ(G d k) = dde, consequently, there exist graphs G such that χc(G) is as close to χ(G) − 1 as you want, but χc(μ(G)) = χ(μ(G)). c © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 63–71, 1999