Circular Coloring and Mycielski Construction

TL;DR

This paper studies the circular chromatic number of graphs obtained via Mycielski construction, proving conditions under which it equals the chromatic number for large parameters and generalizing previous results to Kneser graphs.

Contribution

It establishes new thresholds for when the circular chromatic number equals the chromatic number in Mycielski graphs of Kneser and generalized Kneser graphs.

Findings

For large enough m, the chromatic and circular chromatic numbers coincide.

Existence of a threshold m(n,s,t) for generalized Kneser graphs ensuring equality.

Extension of previous results to broader classes of graphs.

Abstract

In this paper, we investigate circular chromatic number of Mycielski construction of graphs. It was shown in \cite{MR2279672} that $t^{{\rm th}}$ 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 $\chi(M^t(KG(m,n)))=\chi_c(M^t(KG(m,n)))$ where $M^t$ is $t^{{\rm th}}$ Mycielskian. Also, we consider the generalized Kneser graph $KG(m,n,s)$ and show that there exists a threshold $m(n,s,t)$ such that $\chi(M^t(KG(m,n,s)))=\chi_c(M^t(KG(m,n,s)))$ for $m\geq m(n,s,t)$.