Finding long cycles in graphs

TL;DR

This paper investigates methods for detecting long cycles, including Hamiltonian cycles, in graphs using two algorithms: a message passing approach and a Monte Carlo Markov Chain strategy, with focus on non-regular random graphs.

Contribution

It introduces and compares two novel algorithms for finding long cycles in graphs, emphasizing Hamiltonian cycles in non-regular random graphs.

Findings

The message passing algorithm effectively finds long cycles in certain graph classes.

The Monte Carlo approach provides a complementary method with different performance characteristics.

Both algorithms perform well on non-regular random graphs with minimal connectivity of three.

Abstract

We analyze the problem of discovering long cycles inside a graph. We propose and test two algorithms for this task. The first one is based on recent advances in statistical mechanics and relies on a message passing procedure. The second follows a more standard Monte Carlo Markov Chain strategy. Special attention is devoted to Hamiltonian cycles of (non-regular) random graphs of minimal connectivity equal to three.