Document Open Access Logo

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

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

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2024.23.pdf
  • Filesize: 0.79 MB
  • 18 pages

Document Identifiers

Related Versions

Subject Classification

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

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.

Cite As Get BibTex

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

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).

References

  1. Eshrat Arjomandi. An efficient algorithm for colouring the edges of a graph with δ+ 1 colours. INFOR: Information Systems and Operational Research, 20(2):82-101, 1982. Google Scholar
  2. Sepehr Assadi. Faster vizing and near-vizing edge coloring algorithms. CoRR, abs/2405.13371, 2024. URL: https://doi.org/10.48550/arXiv.2405.13371.
  3. Sayan Bhattacharya, Din Carmon, Martín Costa, Shay Solomon, and Tianyi Zhang. Faster (Δ+1)-Edge Coloring: Breaking the m√n Time Barrier. In 65th IEEE Symposium on Foundations of Computer Science (FOCS), 2024. Google Scholar
  4. Sayan Bhattacharya, Martín Costa, Nadav Panski, and Shay Solomon. Arboricity-dependent algorithms for edge coloring. CoRR, abs/2311.08367, 2023. To appear at SWAT'24. URL: https://doi.org/10.48550/arXiv.2311.08367.
  5. Béla Bollobás. Graph theory. Elsevier, 1982. Google Scholar
  6. Yan Cao, Guantao Chen, Suyun Jiang, Huiqing Liu, and Fuliang Lu. Average degrees of edge-chromatic critical graphs. Discrete Mathematics, 342(6):1613-1623, 2019. Google Scholar
  7. Norishige Chiba and Takao Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Comput., 14(1):210-223, 1985. URL: https://doi.org/10.1137/0214017.
  8. Aleksander B. G. Christiansen, Eva Rotenberg, and Juliette Vlieghe. Sparsity-parameterised dynamic edge colouring. CoRR, abs/2311.10616, 2023. To appear at SWAT'24. URL: https://doi.org/10.48550/arXiv.2311.10616.
  9. Aleksander Bjørn Grodt Christiansen. The power of multi-step vizing chains. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 1013-1026. ACM, 2023. Google Scholar
  10. Marek Chrobak and Takao Nishizeki. Improved edge-coloring algorithms for planar graphs. Journal of Algorithms, 11(1):102-116, 1990. Google Scholar
  11. Richard Cole and Łukasz Kowalik. New linear-time algorithms for edge-coloring planar graphs. Algorithmica, 50(3):351-368, 2008. Google Scholar
  12. Stanley Fiorini. Some remarks on a paper by vizing on critical graphs. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 77, pages 475-483. Cambridge University Press, 1975. Google Scholar
  13. Harold N Gabow. Using euler partitions to edge color bipartite multigraphs. International Journal of Computer & Information Sciences, 5(4):345-355, 1976. Google Scholar
  14. Harold N Gabow, Takao Nishizeki, Oded Kariv, Daniel Leven, and Osamu Terada. Algorithms for edge-coloring graphs. Technical Report, 1985. Google Scholar
  15. Dawit Haile. Bounds on the size of critical edge-chromatic graphs. Ars Combinatoria, 53:85-96, 1999. Google Scholar
  16. Ian Holyer. The np-completeness of edge-coloring. SIAM Journal on computing, 10(4):718-720, 1981. Google Scholar
  17. Łukasz Kowalik. Edge-coloring sparse graphs with Δ colors in quasilinear time. arXiv preprint arXiv:2401.13839, 2024. Google Scholar
  18. Jayadev Misra and David Gries. A constructive proof of vizing’s theorem. Information Processing Letters, 41(3):131-133, 1992. Google Scholar
  19. C St JA Nash-Williams. Decomposition of finite graphs into forests. Journal of the London Mathematical Society, 1(1):12-12, 1964. Google Scholar
  20. Daniel P Sanders and Yue Zhao. On the size of edge chromatic critical graphs. Journal of Combinatorial Theory, Series B, 86(2):408-412, 2002. Google Scholar
  21. Corwin Sinnamon. A randomized algorithm for edge-colouring graphs in o(m√n) time. CoRR, abs/1907.03201, 2019. URL: https://arxiv.org/abs/1907.03201.
  22. Vadim G Vizing. On an estimate of the chromatic class of a p-graph. Diskret analiz, 3:25-30, 1964. Google Scholar
  23. Douglas R Woodall. The average degree of an edge-chromatic critical graph ii. Journal of Graph Theory, 56(3):194-218, 2007. Google Scholar
  24. Douglas R Woodall. The average degree of an edge-chromatic critical graph. Discrete mathematics, 308(5-6):803-819, 2008. Google Scholar
  25. Xiao Zhou and Takao Nishizeki. Edge-coloring and f-coloring for various classes of graphs. In Algorithms and Computation: 5th International Symposium, ISAAC'94 Beijing, PR China, August 25-27, 1994 Proceedings 5, pages 199-207. Springer, 1994. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact