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

In the distributed message-passing setting a communication network is represented by a graph whose vertices represent processors that perform local computations and communicate over the edges of the graph. In the distributed edge-coloring problem the processors are required to assign colors to edges, such that all edges incident on the same vertex are assigned distinct colors. The previouslyknown deterministic algorithms for edge-coloring employed at least (2∆− 1) colors, even though any graph admits an edge-coloring with ∆ + 1 colors [36]. Moreover, the previously-known deterministic algorithms that employed at most O(∆) colors required superlogarithmic time [3, 6, 7, 17]. In the current paper we devise deterministic edge-coloring algorithms that employ only ∆ + o(∆) colors, for a very wide family of graphs. Specifically, as long as the arboricity a of the graph is a = O(∆1− ), for a constant > 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 χ in the range [4∆, 2 ∆) · ∆], our χ-edge-coloring algorithm has smaller running time than the best previously-known χ-edge-coloring algorithms. Our algorithms are actually much more general, since edge-coloring is equivalent to vertexcoloring 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. We significantly improve upon previous vertex-coloring of such graphs, and as an implication also obtain the improved edge-coloring algorithms for general 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. We believe that this technique is of independent interest. ∗ Open University of Israel. E-mail: [email protected] ∗∗ Ben-Gurion University of the Negev. Email: [email protected] ∗∗∗ Open University of Israel. Email: [email protected] This research has been supported by the Israeli Academy of Science, grant 724/15. ar X iv :1 61 0. 06 75 9v 1 [ cs .D C ] 2 1 O ct 2 01 6