The diameter of recoloring graphs under a maximum average degree bound

Source abstract

For a graph $G$, we write $\mathrm{mad}(G)$ for its maximum average degree and $\mathrm{diam} G$ for its diameter. Let $R_k(G)$ be the graph whose vertices are the proper colorings of $G$ with $k$ colors, where two colorings are adjacent when they differ at one vertex. Feghali (JCTB, 2021) proved that, for fixed integers $d,k\ge 1$ with $k\ge d+1$ and every $\varepsilon>0$, every $n$-vertex graph $G$ satisfying $\mathrm{mad}(G)\le d-\varepsilon$ has $\mathrm{diam} R_k(G)=O_{d,k,\varepsilon}(n(\log n)^{d-1})$. In this article, we prove that $$\mathrm{diam} R_k(G)=O_{d,k,\varepsilon}\!\left( n(\log n)^{\left\lfloor (d-1)/(k-d)\right\rfloor} \right),$$ which extends the result proved by Feghali directly. The proof uses a partition into independent layers and removes $k-d$ colors at each recursive stage. We also improve the bound on the number of layers and determine the best possible linear coefficient in the forest case. More precisely, for $0<\varepsilon<2$, every $n$-vertex graph $G$ with $\mathrm{mad}(G)\le2-\varepsilon$ satisfies $\mathrm{diam} R_3(G)\leρ_M n$, where $M=\lfloor2/\varepsilon\rfloor$, $ρ_M=\max_{1\le m\le M}D_m/m$, and $D_m$ is the largest diameter of $R_3(T)$ over all trees $T$ on $m$ vertices. Moreover, the coefficient $ρ_M$ is best possible.