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Published online by Cambridge University Press: 14 November 2023
We show that for any $\varepsilon \gt 0$ and
$\Delta \in \mathbb{N}$, there exists
$\alpha \gt 0$ such that for sufficiently large
$n$, every
$n$-vertex graph
$G$ satisfying that
$\delta (G)\geq \varepsilon n$ and
$e(X, Y)\gt 0$ for every pair of disjoint vertex sets
$X, Y\subseteq V(G)$ of size
$\alpha n$ contains all spanning trees with maximum degree at most
$\Delta$. This strengthens a result of Böttcher, Han, Kohayakawa, Montgomery, Parczyk, and Person.
Jie Hu: Supported by Natural Science Foundation of China (12131013, 12161141006). Guanghui Wang: Research supported by Natural Science Foundation of China (12231018) and Young Taishan Scholars probgram of Shandong Province (201909001). Donglei Yang: Supported by the China Post-doctoral Science Foundation (2021T140413), Natural Science Foundation of China (12101365) and Natural ScienceFoundation of Shandong Province (ZR2021QA029).