Circular chromatic number for iterated Mycielski graphs

Abstract For a graph G , let M( G ) denote the Mycielski graph of G . The t th iterated Mycielski graph of G , M t ( G ), is defined recursively by M 0 ( G )= G , and M t ( G )=M(M t −1 ( G )) for t ⩾1. Let χ c ( G ) denote the circular chromatic number of G . We prove two main results: (1) If G has a universal vertex x , then χ c (M( G ))= χ (M( G )) if χ c (G−x)>χ(G)− 1 2 and G is not a star, otherwise χ c ( M (G))=χ( M (G))− 1 2 ; and (2) χ c (M t ( K m ))= χ (M t ( K m )) if m ⩾2 t −1 +2 t −2 and t ⩾2.