A Tight Lower bound on Trees in Graphs

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.