Corr\'adi and Hajnal's theorem for sparse random graphs

TL;DR

This paper extends Corrádi and Hajnal's theorem to sparse random graphs, showing that under certain conditions, large subgraphs contain near-perfect triangle packings, with the result being asymptotically optimal.

Contribution

It introduces a new embedding theorem for regular 3-partite graphs and applies it to establish a sparse random graph analogue of Corrádi and Hajnal's theorem.

Findings

Threshold for p(n) is optimal up to a log factor.

Almost all large subgraphs contain a near-perfect triangle packing.

Introduces a novel embedding theorem for regular 3-partite graphs.

Abstract

In this paper we extend a classical theorem of Corr\'adi and Hajnal into the setting of sparse random graphs. We show that if $p(n) \gg (\log n / n)^{1/2}$, then asymptotically almost surely every subgraph of $G(n,p)$ with minimum degree at least $(2/3 + o(1))np$ contains a triangle packing that covers all but at most $O(p^{-2})$ vertices. Moreover, the assumption on $p$ is optimal up to the $(\log n)^{1/2}$ factor and the presence of the set of $O(p^{-2})$ uncovered vertices is indispensable. The main ingredient in the proof, which might be of independent interest, is an embedding theorem which says that if one imposes certain natural regularity conditions on all three pairs in a balanced 3-partite graph, then this graph contains a perfect triangle packing.