A Tight Lower bound on Trees in Graphs
Chase Wilson
TL;DR
The paper establishes a tight lower bound on the number of injective copies of a t+1-vertex tree T in an n-vertex graph G with average degree d, proving Inj(T,G) ≥ n d(d-1)…(d-t+1) for sufficiently large d. The authors use an entropy-based, random greedy embedding of T into G, carefully bounding the entropy of the embedding and controlling deviations via non-complete partial embeddings and a set of technical lemmas. They show equality characterizations depending on the diameter of T (disjoint unions of K_{d+1} when diam(T)≥3, or d-regular graphs when diam(T)=2) and obtain a polynomial (in t) bound on the required d0(t). The work not only confirms MV’s conjecture and strengthens prior results but also connects to Erdős–Sós and Sidorenko-type questions, with potential applications to counting substructures in graphs and trees more generally.
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.