Circular chromatic number of induced subgraphs of Kneser graphs

TL;DR

This paper explores the conditions under which induced subgraphs of Kneser graphs have equal chromatic and circular chromatic numbers, extending previous results to broader classes of subgraphs.

Contribution

It generalizes existing results to s-stable Kneser graphs and large induced subgraphs of Kneser graphs, establishing when their chromatic and circular chromatic numbers coincide.

Findings

Equal chromatic and circular chromatic numbers for large s-stable Kneser graphs.

Large induced subgraphs of Kneser graphs have equal chromatic and circular chromatic numbers.

Extension of previous results to broader classes of subgraphs.

Abstract

Investigating the equality of the chromatic number and the circular chromatic number of graphs has been an active stream of research for last decades. In this regard, Habolhassan and Zhu [Circular chromatic number of Kneser graphs, Journal of Combinatorial Theory Series B, 2003] proved that if $n$ is sufficiently large with respect to $k$, then the Schrijver graph ${\rm SG}(n,k)$ has the same chromatic and circular chromatic number. Later, Meunier [A topological lower bound for the circular chromatic number of Schrijver graphs, Journal of Graph Theory, 2005] and independently, Simonyi and Tardos [ Local chromatic number, Ky Fan's theorem and circular colorings, Combinatorica, 2006] proved that $\chi({\rm SG}(n,k))=\chi_c({\rm SG}(n,k))$ if $n$ is even. In this paper, we study the circular chromatic number of induced subgraphs of Kneser graphs. In this regard, we shall first generalize the preceding result to $s$-stable Kneser graphs. Furthermore, as a generalization of Hajiabolhassan and Zhu's result, we prove that if $n$ is large enough with respect to $k$, then any sufficiently large induced subgraph of the Kneser graph ${\rm KG}(n,k)$ has the same chromatic number and circular chromatic number.