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 μn(G)=μ(μn−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, χc(μ(G))=χ(μ(G))=4; (4) χc(μ(Kn))=χ(μ(Kn))=n+1 for n⩾3 and χc(μ2(Kn))=χ(μ2(Kn))=n+2 for n⩾4.