We show that every Borel graph $G$ of subexponential growth has a Borel proper edge-coloring with $\Delta(G) + 1$ colors. We deduce this from a stronger result, namely that an $n$-vertex (finite) graph $G$ of subexponential growth can be properly edge-colored using $\Delta(G) + 1$ colors by an $O(\log^\ast n)$-round deterministic distributed algorithm in the $\mathsf{LOCAL}$ model, where the implied constants in the $O(\cdot)$ notation are determined by a bound on the growth rate of $G$.

翻译：我们证明每个次指数增长的Borel图$G$均存在使用$\Delta(G) + 1$种颜色的Borel正常边染色。该结论源自一个更强结果：即对于次指数增长的$n$顶点（有限）图$G$，在$\mathsf{LOCAL}$模型下存在一个$O(\log^\ast n)$轮确定性分布式算法，可使用$\Delta(G) + 1$种颜色对其进行正常边染色，其中$O(\cdot)$记号中的隐含常数由$G$增长率的界决定。