Circular chromatic number for iterated Mycielski graphs

For a graph G, let M(G) denote the Mycielski graph of G. The t-th iterated Mycielski graph of G, M(G), is defined recursively by M0(G) = G and M(G)= M(Mt−1(G)) for t ≥ 1. Let χc(G) denote the circular chromatic number of G. We prove two main results: 1) Assume G has a universal vertex x, then χc(M(G)) = χ(M(G)) if χc(G − x) > χ(G − x) − 1/2 and G is not a star, otherwise χc(M(G)) = χ(M(G)) − 1/2; and 2) if G has 2 t−1 + 2t − 2 universal vertices, then χc(M (G)) = χ(M(G)), improving a result of Hajiabolhassan and Zhu [4]. It is conjectured that χc(M (Kn)) = χ(M (Kn)) for all n ≥ t + 2 [1]. The conjecture is known to be true for t = 1, 2 [1]. A consequence of the result 2) is a shorter proof for the case t = 2.