The circular chromatic number of the Mycielskian ofGdk

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 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 d k . The main result is that χ c (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 χ c (G d k ) = k d and χ(G d k ) = k d , consequently, there exist graphs G such that χ c (G) is as close to χ(G) − 1 as you want, but χ c (µ(G)) = χ(µ(G)).

Packing chromatic number versus chromatic and clique number

“…In this section we also consider the independence number of the Mycielskian. Although the Mycielskian has been investigated by now from many points of view [3,9,19,20,24], it seems that for the independence number only sporadic results were obtained. Setting I(G) to denote the set of independent sets of a graph G (including the empty set), the independence number of the Mycielskian can be described as follows.…”