Strengthening a theorem of Meyniel

TL;DR

This paper improves Meyniel's theorem by proving that the graph of proper 5-colorings of a planar graph, connected via Kempe changes, has a polynomial diameter bound, enhancing understanding of graph coloring reconfigurations.

Contribution

The paper significantly strengthens Meyniel's theorem by establishing a polynomial bound on the diameter of the Kempe change graph for 5-colorings of planar graphs.

Findings

The diameter of the Kempe change graph is polynomial in the number of vertices.

The result applies to all planar graphs with 5-colorings.

It confirms polynomial bounds for reconfiguration sequences in graph coloring.

Abstract

For an integer $k \geq 1$ 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 $G$.