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arXiv · 2510.17736

Spanning trees with large maximum degrees

Jun Yan
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Abstract

The celebrated result of Koml\'os, S\'ark\"ozy, and Szemer\'edi states that for any $\varepsilon>0$, there exists $0<c<1$, such that for all sufficiently large $n$, every $n$-vertex graph $G$ with $\delta(G)\geq(1/2+\varepsilon)n$ contains every $n$-vertex tree with maximum degree at most $cn/\log n$. This is best possible up to the value of $c$. In this paper, we extend this result to trees with higher maximum degrees, and prove that for $\Delta\gg n/\log n$, roughly speaking, $\delta(G)\geq n-n^{1-(1+o(1))\Delta/n}$ is the asymptotically optimal minimum degree condition which guarantees that $G$ contains every $n$-vertex spanning tree with maximum degree at most $\Delta$. We also prove the corresponding statements in the random graph setting.

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Jun Yan. 2025-10-20. Spanning trees with large maximum degrees. https://arxiv.org/abs/2510.17736

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