Quadratic diameter bound for recoloring graphs at the degeneracy threshold

Prove that for every graph G and every integer k satisfying k a0b1a0degen(G)+2, the diameter of the k-recoloring graph C_k(G) is O(n(G)^2).

Background

The paper recalls Cereceda's conjecture concerning the diameter of the ordinary k-recoloring graph C_k(G). Here, degen(G) denotes the degeneracy of G, and n(G) denotes its number of vertices. Connectivity is known when k is at least degen(G)+2, but the best general upper bound cited in the paper is O(n{degen(G)+1}), so the conjectured quadratic bound remains unresolved in general.

The results of the paper concern list-coloring reconfiguration for subcubic and complete multipartite graphs and do not establish this conjecture for arbitrary graphs.

References

Cereceda conjectured that if $k \ge degen(G) + 2 $, the diameter of $C_k(G)$ is $O(n2)$, where $n$ is the number of vertices in $G$.

— Optimal List Recoloring of Subcubic Graphs and Complete Multipartite Graphs  (2501.03748 - Meyer, 7 Jan 2025) in Section 1, Introduction