Kempe changes in degenerate graphs

TL;DR

This paper studies Kempe changes in graph colorings, proving polynomial bounds for equivalence of colorings in graphs with bounded treewidth or maximum degree, and exploring related restrictions.

Contribution

It establishes polynomial bounds on Kempe change sequences for graphs with bounded treewidth or maximum degree, improving understanding of coloring reconfigurations.

Findings

All $k$-colorings are equivalent in $(k-1)$-degenerate graphs.

Polynomial bound of $O(kn^2)$ Kempe changes for graphs with treewidth at most $k-1$.

For graphs with maximum degree $ riangle$, $ riangle$-colorings are equivalent up to $O(n^2)$ Kempe changes, with specific exceptions.

Abstract

We consider Kempe changes on the $k$-colorings of a graph on $n$ vertices. If the graph is $(k-1)$-degenerate, then all its $k$-colorings are equivalent up to Kempe changes. However, the sequence between two $k$-colorings that arises from the proof may be exponential in the number of vertices. An intriguing open question is whether it can be turned polynomial. We prove this to be possible under the stronger assumption that the graph has treewidth at most $k-1$. Namely, any two $k$-colorings are equivalent up to $O(kn^2)$ Kempe changes. We investigate other restrictions (list coloring, bounded maximum average degree, degree bounds). As a main result, we derive that given an $n$-vertex graph with maximum degree $\Delta$, the $\Delta$-colorings are all equivalent up to $O(n^2)$ Kempe changes, unless $\Delta = 3$ and some connected component is a 3-prism.