Round and Communication Efficient Graph Coloring

In the context of communication complexity, we explore protocols for graph coloring, focusing 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 [Distributed Computing 2025], who achieved the same communication cost but required $O(n)$ rounds in expectation, thereby making a significant reduction in the round complexity. Our second contribution is a deterministic protocol to compute a $(2\Delta - 1)$-edge coloring of $G$, which maintains the same $O(n)$ bits of communication and uses only $O(1)$ rounds. We complement the result with a tight $\Omega(n)$-bit lower bound on the communication complexity of the $(2\Delta-1)$-edge coloring problem, while a similar $\Omega(n)$ lower bound for the $(\Delta+1)$-vertex coloring problem has been established by Flin and Mittal [Distributed Computing 2025]. Our result implies a space lower bound of $\Omega(n)$ bits for $(2\Delta - 1)$-edge coloring in the $W$-streaming model, which is the first non-trivial space lower bound for edge coloring in the $W$-streaming model.