A Tight Lower Bound for 3-Coloring Grids in the Online-LOCAL Model

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}