(2∆− 1)-Edge-Coloring is Much Easier than Maximal Matching in Distributed Setting

Graph coloring is a central problem in distributed computing. Both vertexand edge-coloring problems have been extensively studied in this context. In this paper we show that a (2∆ − 1)-edge-coloring can be computed in time smaller than log n for any > 0, specifically, in e √ log logn) rounds. This establishes a separation between the (2∆− 1)-edge-coloring and the Maximal Matching problems, as the latter is known to require Ω( √ log n) time [15]. No such separation is currently known between the (∆+1)-vertex-coloring and the Maximal Independent Set problems. We also devise a (1 + )∆-edge-coloring algorithm for an arbitrarily small constant > 0. This result applies whenever ∆ ≥ ∆ , for some constant ∆ which depends on . The running time of this algorithm is O(log∗∆ · max(1, logn ∆1−o(1) )). The current state-of-the-art is a recent O(log n)-time algorithm by Chung, Pettie and Su (PODC’14) [9]. Similarly to our algorithm, the latter algorithm also assumes ∆ ≥ ∆ for ∆ as above. A much earlier logarithmic-time algorithm by Dubhashi, Grable and Panconesi (ESA’95) [11] assumed ∆ ≥ (log n). For ∆ = (log n) the running time of our algorithm is only O(log∗ n). This constitutes a drastic improvement of the previous logarithmic bound [11, 9]. Our results for (2∆− 1)-edge-coloring follow, in fact, from our far more general results concerning (1− )-locally-sparse graphs. Specifically, we devise a (∆ + 1)-vertex coloring algorithm for (1− )-locally sparse graphs that runs in O(log∗∆+log(1/ )) rounds for any > 0, provided that ∆ = (log n). As a result, we conclude that the (∆ + 1)-vertex coloring problem for (1− )-locally sparse graphs can be solved in O(log(1/ )) + e √ log logn) time. Both these results imply our result about (2∆ − 1)-edge-coloring, because (2∆ − 1)-edgecoloring reduces to (∆ + 1)-vertex-coloring of the line graph of the original graph, and because line graphs are 1/2-locally-sparse. ∗Contact author. Address: 2260 Hayward, Department of EECS, University of Michigan, Ann Arbor, MI 48109. Email: [email protected]. Telephone number: +1 734-680-3514.