Embedding trees using minimum and maximum degree conditions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908720345120768 |
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| author | Pokrovskiy, Alexey Versteegen, Leo Williams, Ella |
| author_facet | Pokrovskiy, Alexey Versteegen, Leo Williams, Ella |
| contents | A variant of the Erdős-Sós conjecture, posed by Havet, Reed, Stein and Wood, states that every graph with minimum degree at least $\lfloor 2k/3 \rfloor$ and maximum degree at least $k$ contains a copy of every tree with $k$ edges. Both degree bounds are best possible. We confirm this conjecture for large trees with bounded maximum degree, by proving that for all $Δ\in \mathbb{N}$ and sufficiently large $k\in \mathbb{N}$, every graph $G$ with $δ(G)\geq \lfloor 2k/3 \rfloor$ and $Δ(G)\geq k$ contains a copy of every tree $T$ with $k$ edges and $Δ(T)\leq Δ$. We also prove similar results where alternative degree conditions are considered. For the same class of trees, this verifies exactly a related conjecture of Besomi, Pavez-Signé and Stein, and provides asymptotic confirmations of two others. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16799 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Embedding trees using minimum and maximum degree conditions Pokrovskiy, Alexey Versteegen, Leo Williams, Ella Combinatorics A variant of the Erdős-Sós conjecture, posed by Havet, Reed, Stein and Wood, states that every graph with minimum degree at least $\lfloor 2k/3 \rfloor$ and maximum degree at least $k$ contains a copy of every tree with $k$ edges. Both degree bounds are best possible. We confirm this conjecture for large trees with bounded maximum degree, by proving that for all $Δ\in \mathbb{N}$ and sufficiently large $k\in \mathbb{N}$, every graph $G$ with $δ(G)\geq \lfloor 2k/3 \rfloor$ and $Δ(G)\geq k$ contains a copy of every tree $T$ with $k$ edges and $Δ(T)\leq Δ$. We also prove similar results where alternative degree conditions are considered. For the same class of trees, this verifies exactly a related conjecture of Besomi, Pavez-Signé and Stein, and provides asymptotic confirmations of two others. |
| title | Embedding trees using minimum and maximum degree conditions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.16799 |