# Distributed CONGEST Algorithm for Finding Hamiltonian Paths in Dirac Graphs and Generalizations

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.

• 1 publication
• 7 publications
• 11 publications
research
05/17/2018

### A Distributed Algorithm for Finding Hamiltonian Cycles in Random Graphs in O(log n) Time

It is known for some time that a random graph G(n,p) contains w.h.p. a H...
research
09/17/2023

### Hamiltonian path and Hamiltonian cycle are solvable in polynomial time in graphs of bounded independence number

A Hamiltonian path (a Hamiltonian cycle) in a graph is a path (a cycle, ...
research
08/03/2022

### Finding a Lower Bound for k-Unbounded Hamiltonian Cycles

Methods to determine the existence of Hamiltonian Cycles in graphs have ...
research
02/27/2019

### Deterministic "Snakes and Ladders" Heuristic for the Hamiltonian Cycle Problem

We present a polynomial complexity, deterministic, heuristic for solving...
research
11/05/2017

### Sparse Kneser graphs are Hamiltonian

For integers k≥ 1 and n≥ 2k+1, the Kneser graph K(n,k) is the graph whos...
research
11/10/2022

### Polyominoes and graphs built from Fibonacci words

We introduce the k-bonacci polyominoes, a new family of polyominoes asso...
research
11/06/2020

### Algorithmic Extensions of Dirac's Theorem

In 1952, Dirac proved the following theorem about long cycles in graphs ...