## Abstract

Graph coloring is a central problem in distributed computing. Both vertex- and edge-coloring problems have been extensively studied in this context. In this paper we show that a (2δ - l)-edge-coloring can be computed in time smaller than log^{∈} n for any ∈ > 0, specifically, in e^{o}(√log log n) rounds. This establishes a separation between the (2δ- l)-edge-coloring and Maximal Matching problems, as the latter is known to require ω(√log n) time [15]. No such separation is currently known between the (δ + l)-vertex-coloring and Maximal Independent Set problems. We devise a (1 + ∈)A-edge-coloring algorithm for an arbitrarily small constant ∈ > 0. This result applies whenever A > Ae, for some constant Ae which depends on e. The running time of this algorithm is O(log∗ δ +log n/δ^{1-0(1)}). A much earlier logarithmic-time algorithm by Dubhashi, Grable and Panconesi [11] assumed δ ≥ (log n)^{1+ω(1)}. For A = (log n) ^{1+n(1)} 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 (2δ - l)-edge-coloring also follows from our more general results concerning (1 - ∈)-locally sparse graphs. Specifically, we devise a (δ + l)-vertex coloring algorithm for (1 - ∈)-locally sparse graphs that runs in O(log∗ δ + log(l/e)) rounds for any ∈ > 0, provided that ∈δ = (log n)^{1+ω(1)}. We conclude that the (δ + 1)-vertex coloring problem for (1 - ∈)-locally sparse graphs can be solved in O(log(l/∈)) + e^{o√loglog n} time. This imply our result about (2δ - l)-edge-coloring, because (2δ - 1)-edge-coloring reduces to (δ + 1)-vertex-coloring of the line graph of the original graph, and because line graphs are (1/2 + o(1))-locally sparse.

Original language | English |
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Pages | 355-370 |

Number of pages | 16 |

State | Published - 1 Jan 2015 |

Event | 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 - San Diego, United States Duration: 4 Jan 2015 → 6 Jan 2015 |

### Conference

Conference | 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 |
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Country/Territory | United States |

City | San Diego |

Period | 4/01/15 → 6/01/15 |

## ASJC Scopus subject areas

- Software
- Mathematics (all)