Circular coloring and Mycielski construction

In this paper, we investigate the circular chromatic number of the iterated Mycielskian of graphs. It was shown by Simonyi and Tardos [G. Simonyi, G. Tardos, Local chromatic number, Ky Fan’s theorem and circular colorings, Combinatorica 26 (5) (2006) 587–626] that the ttth iterate of the Mycielskian of the Kneser graph KG(m,n) has the same circular chromatic number and chromatic number provided that m+tm+t is an even integer. We prove that if mm is large enough, then χ(Mt(KG(m,n)))=χc(Mt(KG(m,n))) where Mt is the ttth iterate of the Mycielskian operator. Also, we consider the generalized Kneser graph KG(m,n,s) and show that there exists a threshold m(n,s,t)m(n,s,t) such that χ(Mt(KG(m,n,s)))=χc(Mt(KG(m,n,s))) for m≥m(n,s,t)m≥m(n,s,t).