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arXiv · 2512.14890

A Tight Lower bound on Trees in Graphs

Chase Wilson
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Abstract

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.

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Chase Wilson. 2025-12-16. A Tight Lower bound on Trees in Graphs. https://arxiv.org/abs/2512.14890

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