Circular chromatic numbers of Mycielskiʹs graphs Original Research Article

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 Mycielskiʹs graphs. In particular, the following results are proved in this paper: (1) for any graph G of chromatic number n, χc(μn−1(G))⩽χ(μn−1(G))−12; (2) if a graph G satisfies χc(G)⩽χ(G)−1d with d=2 or 3, then χc(μ2(G))⩽χ(μ2(G))−1d; (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.