@inproceedings{6ea63c75afdf41cabb6f8b0eba425897,

title = "Distributed CONGEST Algorithm for Finding Hamiltonian Paths in Dirac Graphs and Generalizations",

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 {\~Ω}(n2) rounds, as shown by Bachrach et al. [PODC{\textquoteright}19]. In addition, we consider two generalizations of Dirac graphs: Ore graphs and Rahman-Kaykobad graphs [IPL{\textquoteright}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.",

keywords = "Dirac graphs, graph-algorithms, Hamiltonian Cycle, Hamiltonian Path, Ore graphs, the CONGEST model",

author = "Noy Biton and Reut Levi and Moti Medina",

note = "Publisher Copyright: {\textcopyright} Noy Biton, Reut Levi, and Moti Medina;; 48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023 ; Conference date: 28-08-2023 Through 01-09-2023",

year = "2023",

month = aug,

day = "1",

doi = "10.4230/LIPIcs.MFCS.2023.19",

language = "English",

series = "Leibniz International Proceedings in Informatics, LIPIcs",

publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

editor = "Jerome Leroux and Sylvain Lombardy and David Peleg",

booktitle = "48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023",

address = "Germany",

}