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arXiv · 2502.10147

A Recolouring Version of a Conjecture of Reed

Lucas De Meyer·Clément Legrand-Duchesne·Jared León·Tim Planken·Youri Tamitegama
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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.

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Lucas De Meyer, Clément Legrand-Duchesne, Jared León, Tim Planken, Youri Tamitegama. 2025-02-14. A Recolouring Version of a Conjecture of Reed. https://arxiv.org/abs/2502.10147

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