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arXiv · 2604.25250

Triangle packings in randomly perturbed graphs

Xinbu Cheng·Hong Liu·Lanchao Wang·Zhifei Yan
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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.

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Xinbu Cheng, Hong Liu, Lanchao Wang, Zhifei Yan. 2026-04-28. Triangle packings in randomly perturbed graphs. https://arxiv.org/abs/2604.25250

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