Hamilton cycles in regular graphs perturbed by a random 2-factor

TL;DR

This paper proves that adding a random 2-factor to any regular graph with degree at least 2 results in a Hamiltonian graph with high probability, confirming a conjecture for all degrees.

Contribution

It establishes that the union of a regular graph and a random 2-factor is Hamiltonian for all degrees, resolving a previously open conjecture.

Findings

Union of regular graph and random 2-factor is Hamiltonian with high probability

Confirms conjecture by Draganić and Keevash for all degrees

Applicable to all regular graphs with degree at least 2

Abstract

In this paper, we prove that for each $d \geq 2$, the union of a $d$-regular graph with a uniformly random $2$-factor on the same vertex set is Hamiltonian with high probability. This resolves a conjecture by Dragani\'c and Keevash for all values of $d$.