Sharp threshold for Hamilton cycles in randomly perturbed sparse graphs

Abstract

We determine the sharp threshold for Hamilton cycles in randomly perturbed sparse graphs. For any $\hat{I}\pm =\hat{I}\pm (n)=o(1)$, let $G_{\hat{I}}\pm$ be an $n$-vertex graph with minimum degree $\hat{I}´(G_{\hat{I}}\pm )\ge \hat{I}\pm n$. We prove that if $$p\ge (1+\epsilon )\frac{\log (1/\hat{I}\pm )}{n},$$ then the union $G_{\hat{I}}\pm \cup G(n,p)$ is Hamiltonian asymptotically almost surely. This significantly strengthens a recent result of Hahn-Klimroth, Maesaka, Mogge, Mohr, and Parczyk by improving the leading constant from 6 to the optimal value of 1. Crucially, we show that this bound on $p$ is best possible when $\hat{I}\pm n\to \infty$, thereby establishing the exact probability threshold for Hamiltonicity in this sparse regime. Our proof relies on a robust random expansion lemma, Pósa's booster lemma, and a sprinkling argument.

7 pages