Strengthening a theorem of Meyniel

Abstract

For an integer $k\ge 1$ and a graph $G$, let ${\mathcal{K}}_k(G)$ be the graph that has vertex set all proper $k$-colorings of $G$, and an edge between two vertices $\alpha$ and~$\beta$ whenever the coloring~$\beta$ can be obtained from $\alpha$ by a single Kempe change. A theorem of Meyniel from 1978 states that ${\mathcal{K}}_5(G)$ is connected with diameter $O(5^{|V(G)|})$ for every planar graph $G$. We significantly strengthen this result, by showing that there is a positive constant $c$ such that ${\mathcal{K}}_5(G)$ has diameter $O(|V(G){|}^c)$ for every planar graph $G$.

9 pages, 1 figure