## Abstract

We study the problem of finding a Hamiltonian cycle under the promise that the input graph has a minimum degree of at least n/2, where n denotes the number of vertices in the graph. The classical theorem of Dirac states that such graphs (a.k.a. Dirac graphs) are Hamiltonian, i.e., contain a Hamiltonian cycle. Moreover, finding a Hamiltonian cycle in Dirac graphs can be done in polynomial time in the classical centralized model. This paper presents a randomized distributed CONGEST algorithm that finds w.h.p. a Hamiltonian cycle (as well as maximum matching) within O(log n) rounds under the promise that the input graph is a Dirac graph. This upper bound is in contrast to general graphs in which both the decision and search variants of Hamiltonicity require Ω̃(n^{2}) rounds, as shown by Bachrach et al. [PODC’19]. In addition, we consider two generalizations of Dirac graphs: Ore graphs and Rahman-Kaykobad graphs [IPL’05]. In Ore graphs, the sum of the degrees of every pair of non-adjacent vertices is at least n, and in Rahman-Kaykobad graphs, the sum of the degrees of every pair of non-adjacent vertices plus their distance is at least n + 1. We show how our algorithm for Dirac graphs can be adapted to work for these more general families of graphs.

Original language | English |
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Title of host publication | 48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023 |

Editors | Jerome Leroux, Sylvain Lombardy, David Peleg |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772921 |

DOIs | |

State | Published - Aug 2023 |

Event | 48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023 - Bordeaux, France Duration: 28 Aug 2023 → 1 Sep 2023 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 272 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023 |
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Country/Territory | France |

City | Bordeaux |

Period | 28/08/23 → 1/09/23 |

### Bibliographical note

Publisher Copyright:© Noy Biton, Reut Levi, and Moti Medina;

### Funding

Funding Noy Biton: The author was supported by the Israel Science Foundation under Grant 1867/20. Reut Levi: The author was supported by the Israel Science Foundation under Grant 1867/20. Moti Medina: The author was supported by the Israel Science Foundation under Grant 867/19.

Funders | Funder number |
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Israel Science Foundation | 1867/20, 867/19 |

## Keywords

- Dirac graphs
- Hamiltonian Cycle
- Hamiltonian Path
- Ore graphs
- graph-algorithms
- the CONGEST model