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$G$都具有使用$\Delta(G) + 1$种颜色的Borel真边着色。这一结论可从更强结果推导得出：对于$n$个顶点的次指数增长（有限）图$G$，在$\mathsf{LOCAL}$模型中可通过$O(\log^\ast n)$轮确定性分布式算法，使用$\Delta(G) + 1$种颜色实现真边着色，其中$O(\cdot)$记号隐含的常数由$G$的增长速率上界确定。