Hamilton cycles in weighted Erd\H{o}s-R\'enyi graphs

TL;DR

This paper establishes conditions under which weighted Erdős-Rényi graphs contain multiple edge-disjoint Hamilton cycles and perfect matchings, linking these properties to eigenvalues and minimum degree thresholds.

Contribution

It provides a precise eigenvalue and degree condition characterization for the appearance of multiple Hamilton cycles and perfect matchings in weighted Erdős-Rényi graphs.

Findings

High probability of containing $loor{k/2}$ Hamilton cycles and a perfect matching when minimum degree is at least $k$.

Eigenvalue conditions are crucial for the emergence of these Hamiltonian structures.

Results extend to pseudorandom graphs with specific eigenvalue bounds.

Abstract

Given a symmetric $n\times n$ matrix $P$ with $0 \le P(u, v)\le 1$, we define a random graph $G_{n, P}$ on $[n]$ by independently including any edge $\{u, v\}$ with probability $P(u, v)$. For $k\ge 1$ let $\mathcal{A}_k$ be the property of containing $\lfloor k/2 \rfloor$ Hamilton cycles, and one perfect matching if $k$ is odd, all edge-disjoint. With an eigenvalue condition on $P$, and conditions on its row sums, $G_{n, P}\in \mathcal{A}_k$ happens with high probability if and only if $G_{n, P}$ has minimum degree $k$ whp. We also provide a hitting time version. As a special case, the random graph process on pseudorandom $(n, d, \mu)$-graphs with $\mu \le d(d/n)^\alpha$ for some constant $\alpha > 0$ has property $\mathcal{A}_k$ as soon as it acquires minimum degree $k$ with high probability.