Improved diameter bound for recoloring graphs with maximum-degree-plus-two colors

Establish that for every graph G and every integer k satisfying k a0b1a0Delta(G)+2, the diameter of the k-recoloring graph C_k(G) is at most ceil(3n(G)/2).

Background

For k at least Delta(G)+2, where Delta(G) is the maximum degree, the paper cites a known 2n upper bound on the diameter of C_k(G). Cambie et al. conjectured that this can be reduced to ceil(3n/2).

The paper proves a different list-coloring diameter conjecture for subcubic and complete multipartite graphs and does not resolve this ordinary-coloring conjecture.

References

More recently, Cambie et al. improved this upper bound to $2n$ for $ k \ge \Delta(G) + 2$ and conjectured that it can be further reduced to $\lceil 3n/2 \rceil$.

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