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 t th iterate of the Mycielskian of the Kneser graph KG ( m , n ) has the same circular chromatic number and chromatic number provided that m + t is an even integer. We prove that if m is large enough, then χ ( M t ( KG ( m , n ) ) ) = χ c ( M t ( KG ( m , n ) ) ) where M t is the t th 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 ) such that χ ( M t ( KG ( m , n , s ) ) ) = χ c ( M t ( KG ( m , n , s ) ) ) for m ≥ m ( n , s , t ) .