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arXiv · 2508.18042

Perturbation of dense graphs

Jie Han·Seonghyuk Im·Bin Wang·Junxue Zhang
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Abstract

In the past two decades, various properties of randomly perturbed/augmented (hyper)graphs have been intensively studied, since the model was introduced by Bohman, Frieze and Martin in 2003. The model usually considers a deterministic graph $G$ with minimum degree condition, perturbed/augmented by a binomial random graph $G(n,p)$ on the same vertex set. In this paper, we show that for many problems of finding spanning subgraphs, one can indeed relax the minimum degree condition to a density condition. This includes the embedding problem for $F$-factors when $F$ is not a forest, graphs with bounded maximum degree, $r$-th power of $k$-uniform tight Hamilton cycles for $r,k\ge 2$, and $k$-uniform Hamilton $\ell$-cycles for $\ell\in[2,k-1]$. These results strengthen the results of Balogh, Treglown, and Wagner, of Böttcher, Montgomery, Parczyk, and Person, and of Chang, Han and Thoma.

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Jie Han, Seonghyuk Im, Bin Wang, Junxue Zhang. 2025-08-25. Perturbation of dense graphs. https://arxiv.org/abs/2508.18042

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