Random graph's Hamiltonicity is strongly tied to its minimum degree

TL;DR

This paper demonstrates that in random graphs, the likelihood of lacking a Hamilton cycle closely matches the probability of having a minimum degree less than two, across all edge probabilities.

Contribution

It establishes a precise probabilistic relationship between Hamiltonicity and minimum degree in random graphs, extending understanding of graph properties.

Findings

Probability of no Hamilton cycle equals probability of minimum degree less than 2

Results hold for all edge probability functions p(n)

Analogous result for perfect matchings

Abstract

We show that the probability that a random graph $G\sim G(n,p)$ contains no Hamilton cycle is $(1+o(1))Pr(\delta (G) < 2)$ for all values of $p = p(n)$. We also prove an analogous result for perfect matchings.