Year: 2026
Type: preprint
Abstract: The longstanding Nash-Williams conjecture asserts that every $K_3$-divisible graph $G$ with $δ(G)\ge 3n/4$ admits a triangle decomposition. In the random setting, Frankl and Rödl 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
Source: arXiv (Cornell University)
Authors: Cheng, Xinbu, Liu, Hong, Wang, Lanchao, Yan, Zhifei
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G2485548420, G2765210714
OpenAlex ID: W7158154338
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