Robustness for expander graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914129731649536 |
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| author | Chen, Yaobin Chen, Yu Han, Jie Zhao, Jingwen |
| author_facet | Chen, Yaobin Chen, Yu Han, Jie Zhao, Jingwen |
| contents | We study robust versions of properties of $(n,d,λ)$-graphs, namely, the property of a random sparsification of an $(n,d,λ)$-graph, where each edge is retained with probability $p$ independently. We prove such results for the containment problem of perfect matchings, Hamiltonian cycles, and triangle factors. These results address a series of problems posed by Frieze and Krivelevich.
First we prove that given $γ>0$, for sufficient large $n$, any $(n,d,λ)$-graph $G$ with $λ=o(d)$, $d=Ω(\log n)$ and $p\ge\frac{(1+γ)\log n}{d}$, $G\cap G(n,p)$ contains a Hamiltonian cycle (and thus a perfect matching if $n$ is even) with high probability. This result is asymptotically optimal.
Moreover, we show that for sufficient large $n$, any $(n,d,λ)$-graph $G$ with $λ=o(\frac{d^2}{n})$, $d=Ω(n^{\frac{5}{6}}\log^{\frac{1}{2}}n)$ and $p\gg d^{-1}n^{\frac{1}{3}}\log^{\frac{1}{3}} n$, $G\cap G(n,p)$ contains a triangle factor with high probability. Here, the restrictions on $p$ and $λ$ are asymptotically optimal.
Our proof for the triangle factor problem uses the iterative absorption approach to build a spread measure on the triangle factors, and we also prove and use a coupling result for triangles in the random subgraph of an expander $G$ and the hyperedges in the random subgraph of the triangle-hypergraph of $G$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_00404 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Robustness for expander graphs Chen, Yaobin Chen, Yu Han, Jie Zhao, Jingwen Combinatorics Probability 05C80 05C08, 05C30, 05C48 We study robust versions of properties of $(n,d,λ)$-graphs, namely, the property of a random sparsification of an $(n,d,λ)$-graph, where each edge is retained with probability $p$ independently. We prove such results for the containment problem of perfect matchings, Hamiltonian cycles, and triangle factors. These results address a series of problems posed by Frieze and Krivelevich. First we prove that given $γ>0$, for sufficient large $n$, any $(n,d,λ)$-graph $G$ with $λ=o(d)$, $d=Ω(\log n)$ and $p\ge\frac{(1+γ)\log n}{d}$, $G\cap G(n,p)$ contains a Hamiltonian cycle (and thus a perfect matching if $n$ is even) with high probability. This result is asymptotically optimal. Moreover, we show that for sufficient large $n$, any $(n,d,λ)$-graph $G$ with $λ=o(\frac{d^2}{n})$, $d=Ω(n^{\frac{5}{6}}\log^{\frac{1}{2}}n)$ and $p\gg d^{-1}n^{\frac{1}{3}}\log^{\frac{1}{3}} n$, $G\cap G(n,p)$ contains a triangle factor with high probability. Here, the restrictions on $p$ and $λ$ are asymptotically optimal. Our proof for the triangle factor problem uses the iterative absorption approach to build a spread measure on the triangle factors, and we also prove and use a coupling result for triangles in the random subgraph of an expander $G$ and the hyperedges in the random subgraph of the triangle-hypergraph of $G$. |
| title | Robustness for expander graphs |
| topic | Combinatorics Probability 05C80 05C08, 05C30, 05C48 |
| url | https://arxiv.org/abs/2511.00404 |