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与Suomela（ICALP 2023）从统一视角研究了分布式、顺序、动态及在线场景下图问题的局部性。他们在$\mathsf{Online}$-$\mathsf{LOCAL}$模型中为二分图设计了一种新颖的$O(\log n)$局部性算法，用于实现正确的3-着色。本文通过证明即使在网格上仍成立的紧致$\Omega(\log n)$局部性下界，验证了该算法的最优性。此外，我们进一步证明了环面网格与圆柱网格的3-着色问题存在更高的$\Omega(\sqrt{n})$下界。