We study the communication complexity of $(\Delta + 1)$ vertex coloring, where the edges of an $n$-vertex graph of maximum degree $\Delta$ are partitioned between two players. We provide a randomized protocol which uses $O(n)$ bits of communication and ends with both players knowing the coloring. Combining this with a folklore $\Omega(n)$ lower bound, this settles the randomized communication complexity of $(\Delta + 1)$-coloring up to constant factors.

翻译：我们研究了 $(\Delta + 1)$ 顶点着色的通信复杂度问题，其中最大度为 $\Delta$ 的 $n$ 顶点图的边被划分给两个玩家。我们提出一种随机化协议，该协议使用 $O(n)$ 比特的通信，并以两个玩家均知晓着色方案结束。结合经典的 $\Omega(n)$ 下界，这确定了 $(\Delta + 1)$ 着色的随机化通信复杂度在常数因子内的最优性。