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 $G$.
翻译:对于整数 $\ geq 1 美元 和 图形 G$, 请让 $\ mathcal{ K ⁇ k( G) $ 来表示每张平面图的直径 $( 5)V( G) $ $) 和两个正值 $alpha$ 和 ~\ beta$ 之间的边缘。 1978年的Meyniel 的理论指出, 每张平面图的直径 $( 5) V( G) $) 与 $O ( 5) V( G) $ 相联 。 我们大大加强了这一结果, 我们通过显示正值常值 $ =5 ( G) 表示每张平面图的美元直径为$( ⁇ V ( G) $。