Decentralized Distributed Graph Coloring: Cluster Graphs

Graph coloring is fundamental to distributed computing. We give an ultrafast distributed algorithm for coloring cluster graphs. These graphs are obtained from the underlying communication network by contracting nodes and edges, and they appear frequently as components in the study of distributed algorithms. In particular, we give a $O(\log^* n)$-round algorithm to $\Delta+1$-color cluster graphs of at least polylogarithmic degree. The previous best bound known was $poly(\log n)$ [Flin et.al, SODA'24]. This properly generalizes results in the COONGEST model and shows that distributed graph problems can be quickly solved even when the node itself is decentralized.