Local resilience for squares of almost spanning cycles in sparse random graphs

TL;DR

This paper extends classical results on Hamiltonian squares to sparse random graphs, showing that with high probability, subgraphs with sufficiently high minimum degree contain large squared cycles, nearly matching the deterministic case.

Contribution

It generalizes the Pósa conjecture to sparse random graphs, establishing near-optimal conditions for the existence of squared cycles in such graphs.

Findings

Subgraphs with minimum degree above (2/3 + ε)np contain large squared cycles.

The result is nearly optimal, as below certain thresholds the property fails.

Resilience for spanning squared cycles does not hold in this setting.

Abstract

In 1962, P\'osa conjectured that a graph $G=(V, E)$ contains a square of a Hamiltonian cycle if $\delta(G)\ge 2n/3$. Only more than thirty years later Koml\'os, S\'ark\H{o}zy, and Szemer\'edi proved this conjecture using the so-called Blow-Up Lemma. Here we extend their result to a random graph setting. We show that for every $\epsilon > 0$ and $p=n^{-1/2+\epsilon}$ a.a.s. every subgraph of $G_{n,p}$ with minimum degree at least $(2/3+\epsilon)np$ contains the square of a cycle on $(1-o(1))n$ vertices. This is almost best possible in three ways: (1) for $p\ll n^{-1/2}$ the random graph will not contain any square of a long cycle (2) one cannot hope for a resilience version for the square of a spanning cycle (as deleting all edges in the neighborhood of single vertex destroys this property) and (3) for $c<2/3$ a.a.s. $G_{n,p}$ contains a subgraph with minimum degree at least $cnp$ which does not contain the square of a path on $(1/3+c)n$ vertices.