Deterministic Distributed (Delta + o(Δ))-Edge-Coloring, and Vertex-Coloring of Graphs with Bounded Diversity

We consider coloring problems in the distributed message-passing setting. The previously-known deterministic algorithms for edge-coloring employed at least (2Delta - 1) colors, even though any graph admits an edge-coloring with Delta + 1 colors [V64]. Moreover, the previously-known deterministic algorithms that employed at most O(Delta) colors required superlogarithmic time [B15,BE10,BE11,FHK15]. In the current paper we devise deterministic edge-coloring algorithms that employ only Delta + o(Delta) colors, for a very wide family of graphs. Specifically, as long as the arboricity is a = O(Delta^{1 - \epsilon}), for a constant epsilon > 0, our algorithm computes such a coloring within {polylogarithmic} deterministic time. We also devise significantly improved deterministic edge-coloring algorithms for {general graphs} for a very wide range of parameters. Specifically, for any value $\chi$ in the range [4Delta, 2^{o(log Delta)} \cdot Delta], our \chi-edge-coloring algorithm has smaller running time than the best previously-known \chi-edge-coloring algorithms. Our algorithms are actually much more general, since edge-coloring is equivalent to {vertex-coloring of line graphs.} Our method is applicable to vertex-coloring of the family of graphs with {bounded diversity} that contains line graphs, line graphs of hypergraphs, and many other graphs. Our results are obtained using a novel technique that connects vertices or edges in a certain way that reduces clique size. The resulting structures, which we call {connectors}, can be colored more efficiently than the original graph. Moreover, the color classes constitute simpler subgraphs that can be colored even more efficiently using appropriate connectors. Hence, we recurse until we obtain sufficiently simple structures that are colored directly. We introduce several types of connectors that are useful for various scenarios.