Recently, Akbari, Eslami, Lievonen, Melnyk, S\"{a}rkij\"{a}rvi, and Suomela (ICALP 2023) studied the locality of graph problems in distributed, sequential, dynamic, and online settings from a unified point of view. They designed a novel $O(\log n)$-locality algorithm for proper 3-coloring bipartite graphs in the $\mathsf{Online}$-$\mathsf{LOCAL}$ model. In this work, we show the optimality of the algorithm by demonstrating a tight $\Omega(\log n)$ locality lower bound which holds even on grids. Moreover, we show a higher $\Omega(\sqrt{n})$ lower bound for 3-coloring toroidal and cylindrical grids.

翻译：近期，Akbari、Eslami、Lievonen、Melnyk、Särkijärvi 和 Suomela（ICALP 2023）从统一视角研究了图问题在分布式、顺序、动态和在线环境中的局部性。他们在 $\mathsf{Online}$-$\mathsf{LOCAL}$ 模型中为二部图的恰当三染色设计了一种新颖的 $O(\log n)$-局部性算法。本文通过展示一个即使在网格上也成立的紧的 $\Omega(\log n)$ 局部性下界，证明了该算法的最优性。此外，我们给出了环形网格和圆柱形网格三染色的更高下界 $\Omega(\sqrt{n})$。