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Longest Cycle Above Erdős-Gallai Bound

Authors Fedor V. Fomin , Petr A. Golovach , Danil Sagunov , Kirill Simonov

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Author Details

Fedor V. Fomin
  • Department of Informatics, University of Bergen, Norway
Petr A. Golovach
  • Department of Informatics, University of Bergen, Norway
Danil Sagunov
  • St. Petersburg Department of V.A. Steklov Institute of Mathematics, Russia
Kirill Simonov
  • Algorithms and Complexity Group, TU Wien, Austria

Funding

  • Fomin, Fedor V.: supported by the Research Council of Norway via the project BWCA (grant no. 314528).
  • Golovach, Petr A.: supported by the Research Council of Norway via the project BWCA (grant no. 314528).
  • Sagunov, Danil: supported by Leonhard Euler International Mathematical Institute in Saint Petersburg (agreement no. 075-15-2019-1620).
  • Simonov, Kirill: supported by the Austrian Science Fund (FWF) via project Y1329 (Parameterized Analysis in Artificial Intelligence).

Cite As Get BibTex

Fedor V. Fomin, Petr A. Golovach, Danil Sagunov, and Kirill Simonov. Longest Cycle Above Erdős-Gallai Bound. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ESA.2022.55

Abstract

In 1959, Erdős and Gallai proved that every graph G with average vertex degree ad(G) ≥ 2 contains a cycle of length at least ad(G). We provide an algorithm that for k ≥ 0 in time 2^𝒪(k)⋅n^𝒪(1) decides whether a 2-connected n-vertex graph G contains a cycle of length at least ad(G)+k. This resolves an open problem explicitly mentioned in several papers. The main ingredients of our algorithm are new graph-theoretical results interesting on their own.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Longest path
  • longest cycle
  • fixed-parameter tractability
  • above guarantee parameterization
  • average degree
  • Erdős and Gallai theorem

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References

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