Triangle packings in randomly perturbed graphs

TL;DR

This paper investigates near-perfect triangle packings in randomly perturbed graphs, establishing thresholds for the probability parameter p and introducing a new triangle-weighting lemma.

Contribution

It proves that for certain densities and probabilities, the union of a regular graph and a random graph contains a near-perfect triangle packing, identifying the optimal threshold for p.

Findings

For every d>0 and p>2d/(1+2d), the union contains a near-perfect triangle packing with high probability.

The threshold on p is proven to be optimal for 0<d≤1/2.

Introduces a new triangle-weighting lemma for weighted complete graphs.

Abstract

The longstanding Nash-Williams conjecture asserts that every $K_3$-divisible graph $G$ with $\delta(G)\ge 3n/4$ admits a triangle decomposition. In the random setting, Frankl and R\"odl showed that, with high probability, $G(n,p)$ contains a triangle packing covering all but $o(n^2p)$ edges whenever $p\ge n^{-1/2+\varepsilon}$. In this paper, we study near-perfect triangle packings in randomly perturbed graphs. We prove that for every $d>0$ and every $p>2d/(1+2d)$, if $G_d$ is a $dn$-regular graph on $n$ vertices, then with high probability the union $G_d\cup G(n,p)$ contains a triangle packing covering all but $o(n^2)$ edges. Moreover, this bound on $p$ is best possible for $0<d\le 1/2$, thereby determining the threshold in this range. A key ingredient in the proof is a new triangle-weighting lemma for weighted complete graphs.