# Reconfiguring colorings of graphs with bounded maximum average degree

## TL;DR

This paper investigates the reconfiguration graph of graph colorings, showing that for graphs with bounded maximum average degree, the diameter of the reconfiguration graph grows polynomially with the number of vertices, improving previous bounds.

## Contribution

It proves that for graphs with maximum average degree slightly less than a fixed value, the reconfiguration graph of k-colorings has a diameter bounded by a polynomial function, strengthening existing results.

## Key findings

- Diameter of reconfiguration graph is polynomial in number of vertices.
- Results apply to graphs with maximum average degree less than a threshold.
- Strengthens previous bounds on reconfiguration graph diameters.

## Abstract

The reconfiguration graph $R_k(G)$ for the $k$-colorings of a graph $G$ has as vertex set the set of all possible $k$-colorings of $G$ and two colorings are adjacent if they differ in the color of exactly one vertex of $G$. Let $d, k \geq 1$ be integers such that $k \geq d+1$. We prove that for every $\epsilon > 0$ and every graph $G$ with $n$ vertices and maximum average degree $d - \epsilon$, $R_k(G)$ has diameter $O(n(\log n)^{d - 1})$. This significantly strengthens several existing results.