Recently, \citeauthor*{akbari2021locality}~(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 deterministic algorithm for proper 3-coloring bipartite graphs in the $\mathsf{Online}$-$\mathsf{LOCAL}$ model. In this work, we establish the optimality of the algorithm by showing a \textit{tight} deterministic $\Omega(\log n)$ locality lower bound, which holds even on grids. To complement this result, we have the following additional results: \begin{enumerate} \item We show a higher and {tight} $\Omega(\sqrt{n})$ lower bound for 3-coloring toroidal and cylindrical grids. \item Considering the generalization of $3$-coloring bipartite graphs to $(k+1)$-coloring $k$-partite graphs, %where $k \geq 2$ is a constant, we show that the problem also has $O(\log n)$ locality when the input is a $k$-partite graph that admits a \emph{locally inferable unique coloring}. This special class of $k$-partite graphs covers several fundamental graph classes such as $k$-trees and triangular grids. Moreover, for this special class of graphs, we show a {tight} $\Omega(\log n)$ locality lower bound. \item For general $k$-partite graphs with $k \geq 3$, we prove that the problem of $(2k-2)$-coloring $k$-partite graphs exhibits a locality of $\Omega(n)$ in the $\onlineLOCAL$ model, matching the round complexity of the same problem in the $\LOCAL$ model recently shown by \citeauthor*{coiteux2023no}~(STOC 2024). Consequently, the problem of $(k+1)$-coloring $k$-partite graphs admits a locality lower bound of $\Omega(n)$ when $k\geq 3$, contrasting sharply with the $\Theta(\log n)$ locality for the case of $k=2$. \end{enumerate}

翻译：近期，\citeauthor*{akbari2021locality}（ICALP 2023）从统一视角研究了分布式、顺序、动态和在线设置下图问题的局域性。他们为$\mathsf{Online}$-$\mathsf{LOCAL}$模型中的二部图设计了一种新颖的$O(\log n)$局部确定性算法，用于正确3-着色。本文通过证明一个\textit{紧}的确定性$\Omega(\log n)$局域下界（该下界即使在网格上也成立）来确立该算法的最优性。作为补充，我们还有以下额外结果：\begin{enumerate} \item 我们证明了环面和圆柱网格的3-着色问题具有更高的\textit{紧}下界$\Omega(\sqrt{n})$。\item 考虑将二部图的3-着色推广到$k$部图的$(k+1)$-着色（其中$k \geq 2$为常数），我们证明当输入图是允许\emph{局部可推断唯一着色}的$k$部图时，该问题也具有$O(\log n)$局域性。这类特殊的$k$部图涵盖了若干基础图类，如$k$-树和三角网格。此外，针对此类特殊图类，我们证明了一个\textit{紧}的$\Omega(\log n)$局域下界。\item 对于$k \geq 3$的一般$k$部图，我们证明$k$部图的$(2k-2)$-着色问题在$\onlineLOCAL$模型中具有$\Omega(n)$的局域性，这与\citeauthor*{coiteux2023no}（STOC 2024）近期证实的同一问题在$\LOCAL$模型中的轮复杂度相匹配。由此，当$k\geq 3$时，$k$部图的$(k+1)$-着色问题的局域下界为$\Omega(n)$，这与$k=2$时的$\Theta(\log n)$局域性形成鲜明对比。\end{enumerate}