The circular chromatic number of the Mycielskian of G(k)(d)

Abstract

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 mu(G), we now call the Mycielskian of G, which has the same clique number as G and whose chromatic number equals chi(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(k)(d). The main result is that chi(c)(mu(G(k)(d))) = chi(mu(G(k)(d))), which settles a problem raised in [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear, and X. Zhu, to appear]. As chi(c)(G(k)(d)) = k/d and chi(G(k)(d)) = [k/d], consequently, there exist graphs G such that chi(c)(G) is as close to chi(G) - 1 as you want, but chi(c)(mu(G)) = chi(mu(G)). (C) 1999 John Wiley & Sons, Inc.