Reconfiguring 10-Colourings of Planar Graphs

Abstract

Listen

Translate article

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 $$\ell = 2k$$ .