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Density-Sensitive Algorithms for (Δ + 1)-Edge Coloring

Authors Sayan Bhattacharya, Martín Costa, Nadav Panski, Shay Solomon

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Author Details

Sayan Bhattacharya
  • University of Warwick, UK
Martín Costa
  • University of Warwick, UK
Nadav Panski
  • Tel Aviv University, Israel
Shay Solomon
  • Tel Aviv University, Israel

Funding

Shay Solomon is funded by the European Union (ERC, DynOpt, 101043159). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. Shay Solomon and Nadav Panski are supported by the Israel Science Foundation (ISF) grant No.1991/1. Shay Solomon is also supported by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel, and the United States National Science Foundation (NSF).

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Sayan Bhattacharya, Martín Costa, Nadav Panski, and Shay Solomon. Density-Sensitive Algorithms for (Δ + 1)-Edge Coloring. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.23

Abstract

Vizing’s theorem asserts the existence of a (Δ+1)-edge coloring for any graph G, where Δ = Δ(G) denotes the maximum degree of G. Several polynomial time (Δ+1)-edge coloring algorithms are known, and the state-of-the-art running time (up to polylogarithmic factors) is Õ(min{m √n, m Δ}), by Gabow, Nishizeki, Kariv, Leven and Terada from 1985, where n and m denote the number of vertices and edges in the graph, respectively. Recently, Sinnamon shaved off a polylog(n) factor from the time bound of Gabow et al.

The arboricity α = α(G) of a graph G is the minimum number of edge-disjoint forests into which its edge set can be partitioned, and it is a measure of the graph’s "uniform density". While α ≤ Δ in any graph, many natural and real-world graphs exhibit a significant separation between α and Δ.

In this work we design a (Δ+1)-edge coloring algorithm with a running time of Õ(min{m √n, m Δ})⋅ α/Δ, thus improving the longstanding time barrier by a factor of α/Δ. In particular, we achieve a near-linear runtime for bounded arboricity graphs (i.e., α = Õ(1)) as well as when α = Õ(Δ/√n). Our algorithm builds on Gabow et al.’s and Sinnamon’s algorithms, and can be viewed as a density-sensitive refinement of them.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Graph Algorithms
  • Edge Coloring
  • Arboricity

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References

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