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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2512.14890 |
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| _version_ | 1866908717555908608 |
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| author | Wilson, Chase |
| author_facet | Wilson, Chase |
| contents | Mubayi and Verstraete conjectured that if $T$ is a tree on $t + 1$ vertices, then any $n$-vertex graph $G$ with average degree $d$ contains at least \[ n d(d - 1) \cdots (d - t + 1) \] labeled copies of $T$ as long as $d$ is sufficiently large compared to $t$. We prove this is true and show that when the diameter of $T$ is at least $3$, equality holds iff $G$ is the disjoint union of cliques of size $d + 1$. When the diameter is $2$, equality holds iff $G$ is $d$-regular. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14890 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Tight Lower bound on Trees in Graphs Wilson, Chase Combinatorics Mubayi and Verstraete conjectured that if $T$ is a tree on $t + 1$ vertices, then any $n$-vertex graph $G$ with average degree $d$ contains at least \[ n d(d - 1) \cdots (d - t + 1) \] labeled copies of $T$ as long as $d$ is sufficiently large compared to $t$. We prove this is true and show that when the diameter of $T$ is at least $3$, equality holds iff $G$ is the disjoint union of cliques of size $d + 1$. When the diameter is $2$, equality holds iff $G$ is $d$-regular. |
| title | A Tight Lower bound on Trees in Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.14890 |