Pancyclic subgraphs of random graphs

TL;DR

This paper proves that for random graphs with edge probability significantly greater than n^{-1/2}, any sufficiently dense Hamiltonian subgraph is pancyclic, extending classical theorems and establishing tight bounds.

Contribution

It establishes the resilience of pancyclicity in random graphs for p much larger than n^{-1/2}, with optimal bounds on edge density and probability range.

Findings

Random graphs with p bc n^{-1/2} are resiliently pancyclic.

Hamiltonian subgraphs with more than half the expected edges are pancyclic.

The bounds on p and edge proportion are shown to be tight.

Abstract

An $n$-vertex graph is called pancyclic if it contains a cycle of length $t$ for all $3 \leq t \leq n$. In this paper, we study pancyclicity of random graphs in the context of resilience, and prove that if $p \gg n^{-1/2}$, then the random graph $G(n,p)$ a.a.s. satisfies the following property: Every Hamiltonian subgraph of $G(n,p)$ with more than $(1/2 + o(1)){n \choose 2}p$ edges is pancyclic. This result is best possible in two ways. First, the range of $p$ is asymptotically tight; second, the proportion 1/2 of edges cannot be reduced. Our theorem extends a classical theorem of Bondy, and is closely related to a recent work of Krivelevich, Lee, and Sudakov. The proof uses a recent result of Schacht (also independently obtained by Conlon and Gowers).