A Recolouring Version of a Conjecture of Reed

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 $\epsilon$ such that all graphs $G$ are $k$-recolourable for all $k\ge \lceil \epsilon \omega (G)+(1-\epsilon )(\Delta (G)+1)\rceil$. For general graphs, an existing construction of a frozen colouring shows that $\epsilon \le 1/3$. 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 $\epsilon =1/3$. For triangle-free graphs, we give a construction of frozen colourings that shows that $\epsilon \le 4/9$, and prove that it is also optimal. In the special case of odd-hole-free graphs, we show that $\epsilon =1/2$, and that this is tight up to one colour.

17 pages, 8 figures