Triangle packings in randomly perturbed graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910172974153728 |
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| author | Cheng, Xinbu Liu, Hong Wang, Lanchao Yan, Zhifei |
| author_facet | Cheng, Xinbu Liu, Hong Wang, Lanchao Yan, Zhifei |
| contents | 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 $0<d\le 1/2$, thereby determining the threshold in this range. A key ingredient in the proof is a new triangle-weighting lemma for weighted complete graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_25250 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Triangle packings in randomly perturbed graphs Cheng, Xinbu Liu, Hong Wang, Lanchao Yan, Zhifei Combinatorics 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 $0<d\le 1/2$, thereby determining the threshold in this range. A key ingredient in the proof is a new triangle-weighting lemma for weighted complete graphs. |
| title | Triangle packings in randomly perturbed graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.25250 |