A Tight Lower bound on Trees in Graphs

TL;DR

This paper proves a conjecture about the minimum number of labeled copies of a tree in a graph with given average degree, establishing exact conditions for when equality holds based on the tree's diameter.

Contribution

It confirms Mubayi and Verstraete's conjecture and characterizes extremal graphs for trees with diameter at least 3 and exactly 2.

Findings

Confirmed the conjecture for large degree d

Characterized extremal graphs for trees with diameter ≥ 3

Characterized extremal graphs for trees with diameter 2

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.