A Recolouring Version of a Conjecture of Reed

TL;DR

This paper explores a recolouring version of Reed's conjecture, establishing optimal bounds for the parameter  in various classes of graphs, and conjectures =1/3 for general graphs.

Contribution

It introduces a recolouring framework related to Reed's conjecture, determines optimal bounds for  in general, triangle-free, and odd-hole-free graphs, and proposes a conjecture for the general case.

Findings

Optimal  bound of 1/3 for general graphs.

Optimal  bound of 4/9 for triangle-free graphs.

Optimal  bound of 1/2 for odd-hole-free graphs.

Abstract

Reed conjectured that the chromatic number of any graph is closer to its clique number than to its maximum degree plus one. We consider a recolouring version of this conjecture, with respect to Kempe changes. Namely, we investigate the largest $\varepsilon$ such that all graphs $G$ are $k$-recolourable for all $k \ge \lceil \varepsilon \omega(G) + (1 -\varepsilon)(\Delta(G)+1) \rceil$. For general graphs, an existing construction of a frozen colouring shows that $\varepsilon \le 1/3$. We show that this construction is optimal in the sense that there are no frozen colourings below that threshold. For this reason, we conjecture that $\varepsilon = 1/3$. For triangle-free graphs, we give a construction of frozen colourings that shows that $\varepsilon \le 4/9$, and prove that it is also optimal. In the special case of odd-hole-free graphs, we show that $\varepsilon = 1/2$, and that this is tight up to one colour.