[Paper Review] The number of trees in a graph

Dhruv Mubayi, Jacques Verstraëte|arXiv (Cornell University)|Nov 23, 2015

Limits and Structures in Graph Theory3 references3 citations

TL;DR

This paper establishes a tight lower bound on the number of isomorphic labeled copies of a tree T with t edges in a graph G of minimum degree at least t and average degree d. Using a novel probability-based induction argument on tree embeddings, it proves that G contains at least $ nd(d - t + 1)^{t - 1} $ such copies, generalizing earlier results for paths and extending Erd\'os and Simonovits' work on tree homomorphisms to isomorphic embeddings with explicit degree-dependent bounds.

ABSTRACT

Let $T$ be a tree with $t$ edges. We show that the number of isomorphic (labeled) copies of $T$ in a graph $G = (V,E)$ of minimum degree at least $t$ is at least \[2|E| \prod_{v \in V} (d(v) - t + 1)^{\frac{(t-1)d(v)}{2|E|}}.\] Consequently, any $n$-vertex graph of average degree $d$ and minimum degree at least $t$ contains at least $$nd(d-t+1)^{t-1}$$ isomorphic (labeled) copies of $T$. This answers a question of Dellamonica et. al. (where the above statement was proved when $T$ is the path with three edges) while extending an old result of Erd\H os and Simonovits.

Motivation & Objective

• To determine a non-trivial lower bound on the number of isomorphic labeled copies of a tree T with t edges in a graph G with minimum degree at least t.
• To generalize a prior result by [3] on paths with three edges to arbitrary trees.
• To extend Erd\'os and Simonovits' asymptotic result on tree walks to a precise, convex-degree-dependent lower bound for isomorphic embeddings.
• To resolve a question posed in [3] about whether a degree-convex lower bound exists for isomorphic tree copies under minimum degree conditions.

Proposed method

• Define a probability measure on isomorphic tree embeddings using a good labeling of T (e.g., via BFS/DFS) and sequential vertex selection avoiding previously used vertices.
• Construct a probability distribution $ \mathbb{P}(\omega) $ for each embedding $ \omega $, inversely proportional to the available choices at each step, with bounds based on vertex degrees minus $ t-1 $.
• Prove via induction on t that the marginal probability $ g_i(v) $ of vertex $ v $ being at position $ i $ in the embedding satisfies $ g_i(v) \geq \frac{d(v)}{nd} $, using symmetry and relabeling arguments.
• Use the inequality of arithmetic and geometric means on the product of probabilities to derive the lower bound on the total number of embeddings.
• Leverage the invariance of the probability function under reverse labeling of the tree (i.e., $ p(\omega) = p(z) $) to handle the last vertex in the embedding sequence.
• Apply Jensen’s inequality to the convex function $ f(x) = (t-1)x \log(x - t + 1) $ to derive the final average-degree-based bound from the vertex-degree-based bound.

Experimental results

Research questions

• RQ1 Can a precise, convex-degree-dependent lower bound be established for the number of isomorphic labeled copies of a tree T in a graph G with minimum degree at least t and average degree d?
• RQ2 Does the bound $ 2|E| \prod_{v \in V} (d(v) - t + 1)^{\frac{(t-1)d(v)}{2|E|}} $ hold for isomorphic tree copies, generalizing the result from [3] on paths?
• RQ3 Is the bound $ nd(d - t + 1)^{t - 1} $ sharp, and does it match the count in extremal graphs like disjoint cliques of size $ d+1 $?
• RQ4 Can the method used for homomorphisms in prior works be adapted to handle isomorphisms, where vertex uniqueness must be enforced during embedding?
• RQ5 Is the marginal probability of a vertex appearing at a given position in a tree embedding bounded below by $ d(v)/nd $, even when the embedding process is non-reversible?

Key findings

• The number of isomorphic labeled copies of a t-edge tree T in a graph G with minimum degree at least t and average degree d is at least $ nd(d - t + 1)^{t - 1} $, as shown in Corollary 2.
• The main result, Theorem 1, establishes a lower bound of $ nd \prod_{v \in V} (d(v) - t + 1)^{\frac{(t-1)d(v)}{nd}} $, which is tighter and more structured than previous bounds.
• The proof uses a novel inductive argument on tree embeddings with a carefully defined probability measure that accounts for vertex availability and avoids overcounting.
• The bound is sharp in the sense that complete graphs (or disjoint cliques of size $ d+1 $) achieve $ nd(d-1)(d-2)\cdots(d-t+1) $, matching the asymptotic form of the bound.
• The paper resolves an open question from [3] by showing that a degree-convex lower bound exists for isomorphic tree copies under minimum degree constraints.
• The method demonstrates that the probability of a vertex appearing at a given position in a tree embedding is bounded below by $ d(v)/nd $, even when the embedding process is non-Markovian due to vertex exclusion.