A polynomial-time algorithm to determine (almost) Hamiltonicity of dense regular graphs

TL;DR

This paper presents a polynomial-time algorithm to detect very long cycles in dense regular graphs, advancing understanding of Hamiltonicity in such graphs with implications for related NP-complete problems.

Contribution

It introduces a novel polynomial-time algorithm for identifying near-Hamiltonian cycles in dense regular graphs, combining extremal graph theory and spectral methods.

Findings

Algorithm detects cycles covering all but a small fraction of vertices

Determines Hamiltonicity in dense regular graphs efficiently

NP-completeness when density or regularity is not assumed

Abstract

We give a polynomial-time algorithm for detecting very long cycles in dense regular graphs. Specifically, we show that, given $\alpha \in (0,1)$, there exists a $c=c(\alpha)$ such that the following holds: there is a polynomial-time algorithm that, given a $D$-regular graph $G$ on $n$ vertices with $D\geq \alpha n$, determines whether $G$ contains a cycle on at least $n - c$ vertices. The problem becomes NP-complete if we drop either the density or the regularity condition. The algorithm combines tools from extremal graph theory and spectral partitioning as well as some further algorithmic ingredients.