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参与者。我们提出了一种随机协议，该协议使用$O(n)$比特的通信，并使得双方最终都知道着色方案。结合已知的$\Omega(n)$下界，这确定了$(\Delta + 1)$着色的随机通信复杂度在常数因子内的精确值。