Towards Cereceda's conjecture for planar graphs
(1810.00731)Abstract
The reconfiguration graph $Rk(G)$ of the $k$-colourings of a graph $G$ has as vertex set the set of all possible $k$-colourings of $G$ and two colourings are adjacent if they differ on the colour of exactly one vertex. Cereceda conjectured ten years ago that, for every $k$-degenerate graph $G$ on $n$ vertices, $R{k+2}(G)$ has diameter $\mathcal{O}({n2})$. The conjecture is wide open, with a best known bound of $\mathcal{O}({kn})$, even for planar graphs. We improve this bound for planar graphs to $2{\mathcal{O}({\sqrt{n}})}$. Our proof can be transformed into an algorithm that runs in $2{\mathcal{O}({\sqrt{n}})}$ time.
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