Reconfiguring 10-Colourings of Planar Graphs

Abstract

Let $k \ge 1$ be an integer. The reconfiguration graph $R_k(G)$ 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 exactly one vertex. A conjecture of Cereceda from 2007 asserts that for every integer $\ell \ge k + 2$ and k-degenerate graph G on n vertices, $R_{\ell }(G)$ has diameter $O(n^2)$. The conjecture has been verified only when $\ell \ge 2k + 1$. We give a simple proof that if G is a planar graph on n vertices, then $R_{10}(G)$ has diameter at most $n(n + 1)/ 2$. Since planar graphs are 5-degenerate, this affirms Cereceda’s conjecture for planar graphs in the case $\ell = 2k$.