Quantitative Small Subgraph Conditioning

TL;DR

This paper refines the small subgraph conditioning method for random regular graphs, providing quantitative convergence rates for Hamiltonicity and establishing distributional and spectral properties of Hamiltonian cycles.

Contribution

It introduces new technical machinery that applies to all values of degree and size, enabling precise estimates and new distributional results for Hamiltonian cycles in random regular graphs.

Findings

Convergence rate of Hamiltonian cycle probability quantified.

Distributional convergence of Hamiltonian cycles established.

Number of Hamiltonian cycles approximated via eigenvalues.

Abstract

We revisit the method of small subgraph conditioning, used to establish that random regular graphs are Hamiltonian a.a.s. We refine this method using new technical machinery for random $d$-regular graphs on $n$ vertices that hold not just asymptotically, but for any values of $d$ and $n$. This lets us estimate how quickly the probability of containing a Hamiltonian cycle converges to 1, and it produces quantitative contiguity results between different models of random regular graphs. These results hold with $d$ held fixed or growing to infinity with $n$. As additional applications, we establish the distributional convergence of the number of Hamiltonian cycles when $d$ grows slowly to infinity, and we prove that the number of Hamiltonian cycles can be approximately computed from the graph's eigenvalues for almost all regular graphs.