In the context of communication complexity, we explore randomized protocols for graph coloring, focusing specifically on the vertex and edge coloring problems in $n$-vertex graphs $G$ with a maximum degree $\Delta$. We consider a scenario where the edges of $G$ are partitioned between two players. Our first contribution is a randomized protocol that efficiently finds a $(\Delta + 1)$-vertex coloring of $G$, utilizing $O(n)$ bits of communication in expectation and completing in $O(\log \log n \cdot \log \Delta)$ rounds in the worst case. This advancement represents a significant improvement over the work of Flin and Mittal [PODC 2024], who achieved the same communication cost but required $O(n)$ rounds in expectation, thereby making a significant reduction in the round complexity. We also present a randomized protocol for a $(2\Delta - 1)$-edge coloring of $G$, which maintains the same $O(n)$ bits of communication in expectation over $O(\log^\ast \Delta)$ rounds in the worst case. We complement the result with a tight $\Omega(n)$-bit lower bound on the communication complexity of the $(2\Delta-1)$-edge coloring, while a similar $\Omega(n)$ lower bound for the $(\Delta+1)$-vertex coloring has been established by Flin and Mittal [PODC 2024].

翻译：在通信复杂性的背景下，我们研究了图着色的随机化协议，特别关注于最大度为 $\Delta$ 的 $n$ 顶点图 $G$ 的顶点着色和边着色问题。我们考虑的场景是 $G$ 的边被划分给两个参与者。我们的第一个贡献是一个随机化协议，它能高效地找到 $G$ 的 $(\Delta + 1)$-顶点着色，期望通信复杂度为 $O(n)$ 比特，且在最坏情况下于 $O(\log \log n \cdot \log \Delta)$ 轮内完成。这一进展相较于 Flin 和 Mittal [PODC 2024] 的工作有显著改进，他们虽然达到了相同的通信开销，但期望轮数为 $O(n)$，从而我们在轮复杂度上实现了显著降低。我们还提出了一个用于 $G$ 的 $(2\Delta - 1)$-边着色的随机化协议，它在最坏情况下于 $O(\log^\ast \Delta)$ 轮内保持期望通信复杂度为 $O(n)$ 比特。我们通过一个紧的 $\Omega(n)$ 比特下界来补充该结果，该下界针对 $(2\Delta-1)$-边着色的通信复杂性，而 Flin 和 Mittal [PODC 2024] 已为 $(\Delta+1)$-顶点着色建立了类似的 $\Omega(n)$ 下界。