2025/02/14 by De Meyer, Lucas, Legrand-Duchesne, Clément, León, Jared +2 · 1 citation
#05C15 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2502.10147
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 ε such that all graphs G are k-recolourable for all k ≥ \lceil ε ω(G) + (1 -ε)(Δ(G)+1) \rceil. For general graphs, an existing construction of a frozen colouring shows that ε ≤ 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 ε = 1/3. For triangle-free graphs, we give a construction of frozen colourings that shows that ε ≤ 4/9, and prove that it is also optimal. In the special case of odd-hole-free graphs, we show that ε = 1/2, and that this is tight up to one colour.