## Abstract

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 (2Δ - 1) colors, even though any graph admits an edge-coloring with Δ + 1 colors [36]. Moreover, the previously-known deterministic algorithms that employed at most O(Δ) colors required superlogarithmic time [3, 6, 7, 17]. In the current paper we devise deterministic edge-coloring algorithms that employ only Δ + o(Δ) colors, for a very wide family of graphs. Specifically, as long as the arboricity a of the graph is a = O(Δ^{1-ϵ}), for a constant ϵ > 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 κ in the range [4Δ, 2^{o(log Δ)} · Δ], our κ-edge-coloring algorithm has smaller running time than the best previously-known κ-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 vertexcoloring 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. We introduce several types of connectors that are useful for various scenarios. We believe that this technique is of independent interest.

Original language | English |
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Title of host publication | PODC 2017 - Proceedings of the ACM Symposium on Principles of Distributed Computing |

Publisher | Association for Computing Machinery |

Pages | 175-184 |

Number of pages | 10 |

ISBN (Electronic) | 9781450349925 |

DOIs | |

State | Published - 26 Jul 2017 |

Event | 36th ACM Symposium on Principles of Distributed Computing, PODC 2017 - Washington, United States Duration: 25 Jul 2017 → 27 Jul 2017 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Principles of Distributed Computing |
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Volume | Part F129314 |

### Conference

Conference | 36th ACM Symposium on Principles of Distributed Computing, PODC 2017 |
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Country/Territory | United States |

City | Washington |

Period | 25/07/17 → 27/07/17 |

### Bibliographical note

Funding Information:* This research has been supported by ISF grant No. 724/15 and by the Open University of Israel research fund. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. PODC ’17, July 25-27, 2017, Washington, DC, USA © 2017 Association for Computing Machinery. ACM ISBN 978-1-4503-4992-5/17/07...$15.00 https://doi.org/10.1145/3087801.3087812

Publisher Copyright:

© 2017 Association for Computing Machinery.

## Keywords

- Clique decomposition
- Distributed coloring
- Network partition