A robust Corr\'adi--Hajnal Theorem

TL;DR

This paper extends classical theorems on triangle factors in dense graphs to a probabilistic setting, showing that a random sparsification of such graphs still likely contains a triangle factor under certain conditions.

Contribution

It establishes a probabilistic version of the Corrádi–Hajnal theorem, identifying conditions on edge probability and minimum degree for the existence of triangle factors in sparse random subgraphs.

Findings

High probability existence of triangle factors in sparse random subgraphs

Tight bounds on minimum degree and probability conditions

Lower bounds on the number of triangle factors in dense graphs

Abstract

For a graph $G$ and $p\in[0,1]$, we denote by $G_p$ the random sparsification of $G$ obtained by keeping each edge of $G$ independently, with probability $p$. We show that there exists a $C>0$ such that if $p\geq C(\log n)^{1/3}n^{-2/3}$ and $G$ is an $n$-vertex graph with $n\in 3\mathbb{N}$ and $\delta(G)\geq \tfrac{2n}{3}$, then with high probability $G_p$ contains a triangle factor. Both the minimum degree condition and the probability condition, up to the choice of $C$, are tight. Our result can be viewed as a common strengthening of the seminal theorems of Corr\'adi and Hajnal, which deals with the extremal minimum degree condition for containing triangle factors (corresponding to $p=1$ in our result), and Johansson, Kahn and Vu, which deals with the threshold for the appearance of a triangle factor in $G(n,p)$ (corresponding to $G=K_n$ in our result). It also implies a lower bound on the number of triangle factors in graphs with minimum degree at least $\tfrac{2n}{3}$ which gets close to the truth.