(2∆− 1)-Edge-Coloring is Much Easier than Maximal Matching in the 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 Maximal Matching problems, as the latter is known to require Ω( √ logn) time [15]. No such separation is currently known between the (∆ + 1)-vertex-coloring and Maximal Independent Set problems. We 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∗∆ + logn ∆1−o(1) ). A much earlier logarithmic-time algorithm by Dubhashi, Grable and Panconesi [11] assumed ∆ ≥ (logn). For ∆ = (logn) 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 also follows from our 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 ∆ = (logn). We conclude that the (∆ + 1)-vertex coloring problem for (1 − )-locally sparse graphs can be solved in O(log(1/ )) + e √ log logn) time. This imply our result about (2∆−1)-edge-coloring, because (2∆−1)-edge-coloring reduces to (∆+1)-vertexcoloring of the line graph of the original graph, and because line graphs are (1/2 + o(1))-locally sparse. ∗This research has been supported by the Israeli Academy of Science, grant 593/11, and by the Binational Science Foundation, grant 2008390. In addition, this research has been supported by the Lynn and William Frankel Center for Computer Science. †Supported by NSF grants CCF-1217338 and CNS1318294. This research was performed partly at the Center for Massive Data Algorithmics (MADALGO) at Aarhus University, which is supported by Danish National Research Foundation grant DNRF84.