Circular chromatic numbers of Mycielski's graphs

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. Let μ n ( G)= μ( μ n−1 ( G)) for n⩾2. This paper investigates the circular chromatic numbers of Mycielski's graphs. In particular, the following results are proved in this paper: (1) for any graph G of chromatic number n, χ c (μ n−1(G))⩽χ(μ n−1(G))− 1 2 ; (2) if a graph G satisfies χ c (G)⩽χ(G)− 1 d with d=2 or 3, then χ c (μ 2(G))⩽χ(μ 2(G))− 1 d ; (3) for any graph G of chromatic number 3, χ c( μ( G))= χ( μ( G))=4; (4) χ c( μ( K n ))= χ( μ( K n ))= n+1 for n⩾3 and χ c( μ 2( K n ))= χ( μ 2( K n ))= n+2 for n⩾4.