A Generalization of Kneser's Conjecture

TL;DR

This paper explores the star-free chromatic number of Kneser graphs, establishing bounds and exact values in certain cases, and proposes a conjecture relating it to the chromatic number.

Contribution

It provides new bounds and exact values for the star-free chromatic number of Kneser graphs, extending understanding of their coloring properties.

Findings

Established that hi_s(KG(n,k))  hi(KG(n,k)) in general.

Proved hi_s(KG(n,k))=2hi(KG(n,k))-2 for n  /3 k.

Proposed a conjecture that hi_s(KG(n,k))=2hi(KG(n,k))-2 for all n  2k  4.

Abstract

We investigate some coloring properties of Kneser graphs. A star-free coloring is a proper coloring $c:V(G)\to \Bbb{N}$ such that no path with three vertices may be colored with just two consecutive numbers. The minimum positive integer $t$ for which there exists a star-free coloring $c: V(G) \to \{1,2,..., t\}$ is called the star-free chromatic number of $G$ and denoted by $\chi_s(G)$. In view of Tucker-Ky Fan's lemma, we show that for any Kneser graph ${\rm KG}(n,k)$ we have $\chi_s({\rm KG}(n,k))\geq \max\{2\chi({\rm KG}(n,k))-10, \chi({\rm KG}(n,k))\}$ where $n\geq 2k \geq 4$. Moreover, we show that $\chi_s({\rm KG}(n,k))=2\chi({\rm KG}(n,k))-2=2n-4k+2$ provided that $n \leq {8\over 3}k$. This gives a partial answer to a conjecture of [12]. Also, we conjecture that for any positive integers $n\geq 2k \geq 4$ we have $\chi_s({\rm KG}(n,k))= 2\chi({\rm KG}(n,k))-2$.