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 $d,k\ge 1$ be integers such that $k\ge d+1$. We prove that for every $ϵ>0$ and every graph G with n vertices and maximum average degree $d-ϵ$, $R_k(G)$ has diameter $O(n{(\log n)}^{d-1})$. This significantly strengthens several existing results.

中文翻译：

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用有界最大平均度重新配置图的颜色

重新配置图 ${\mathrm{[R}}_{ķ}（G）$用于ķ的曲线图的-colorings ģ具有作为顶点设定的设定的所有可能的ķ的-colorings ģ和两个着色剂，如果他们在恰好一个顶点的颜色不同相邻ģ。让$d，ķ\ge \mathrm{1个}$ 是这样的整数 $ķ\ge d+\mathrm{1个}$。我们证明$ϵ>0$每个图G都有n个顶点和最大平均度$d-ϵ$， ${\mathrm{[R}}_{ķ}（G）$ 有直径 $Ø（ñ{（\mathrm{日志}ñ）}^{d-\mathrm{1个}}）$。这大大加强了几个现有的结果。