The circular chromatic number of the Mycielskian of G d k

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 G dk. The main result is that χc(μ(G dk)) = χ(μ(G dk)), which settles a problem raised in [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear, and X. Zhu, to appear]. As χc(G dk) = ${k}\over{d}$ and χ(G dk) = $\lceil {{k}\over{d}} \rceil$, 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)). © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 6371, 1999