When Hamilton circuits generate the cycle space of a random graph

TL;DR

This paper investigates the conditions under which Hamilton circuits generate the entire cycle space in random graphs, revealing that a certain edge probability threshold ensures this property with high probability, which differs from the Hamiltonicity threshold.

Contribution

The paper establishes a probabilistic threshold for when Hamilton circuits generate the entire cycle space in G(n,p), highlighting a gap from the Hamiltonicity threshold.

Findings

Hamilton circuits generate the full cycle space for p >= n^{-1/2 + eps}

The threshold for this property is higher than the Hamiltonicity threshold

The cycle space generated by Hamilton circuits matches parity constraints depending on n's parity

Abstract

If eps > 0 and p >= n^{-1/2 + eps}, in a binomial random graph G(n,p) a.a.s. the set of cycles which can be constructed as a symmetric difference of Hamilton circuits is as large as parity by itself permits (all cycles if n is odd, all even cycles if n is even). Moreover, every p which ensures the above property a.a.s. must necessarily be such that for any constant c>0, eventually p >= (log n + 2 log log n + c)/n. So, whatever the smallest sufficient p for an a.a.s. Hamilton-generated cycle space might be, it does not coincide with the threshold for hamiltonicity of G(n,p).