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Anticoncentration of random spanning trees in graphs with large minimum degree

Veronica Bitonti, Lukas Michel, Alex Scott

Abstract

A classical result by Otter shows that the complete graph has an exponential number of non-isomorphic spanning trees. This was recently extended by Lee to every almost regular graph of sufficiently large degree. In this paper, we consider graphs of large minimum degree. We show that every connected graph $G$ with $n$ vertices and minimum degree $d$ has at least $n^{Ω(d)}$ non-isomorphic spanning trees. This is tight up to the constant factor in the exponent. In fact, we prove the following anticoncentration result: if $\mathcal{T}$ is a uniformly random spanning tree of $G$, then for every tree $T$, the probability that $\mathcal{T}$ is isomorphic to $T$ is at most $n^{-Ω(d)}$. This proves a conjecture of Lee in a strong form.

Anticoncentration of random spanning trees in graphs with large minimum degree

Abstract

A classical result by Otter shows that the complete graph has an exponential number of non-isomorphic spanning trees. This was recently extended by Lee to every almost regular graph of sufficiently large degree. In this paper, we consider graphs of large minimum degree. We show that every connected graph with vertices and minimum degree has at least non-isomorphic spanning trees. This is tight up to the constant factor in the exponent. In fact, we prove the following anticoncentration result: if is a uniformly random spanning tree of , then for every tree , the probability that is isomorphic to is at most . This proves a conjecture of Lee in a strong form.
Paper Structure (8 sections, 12 theorems, 15 equations)

This paper contains 8 sections, 12 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Random spanning trees with many leaves
  3. Anticoncentration for random spanning trees
  4. Leaf reconfiguration strategies
  5. A leaf reconfiguration strategy for a random spanning tree
  6. Anticoncentration for random subgraphs of bipartite graphs
  7. Anticoncentration after the leaf reconfiguration
  8. Open problems

Key Result

Theorem 1.1

Let $G$ be a connected graph with $n$ vertices and minimum degree $d$, and let $d(G) \coloneqq \prod_{v \in V(G)} d_G(v)$. Then, the number of spanning trees of $G$ is at least $d(G) \cdot e^{-\mathcal{O}((\log d)^2/d) n}$ and at most $d(G) / (n-1)$.

Theorems & Definitions (15)

  • Theorem 1.1: Kostochka Kostochka
  • Theorem 1.2: Lee Lee
  • Conjecture 1.3: Lee Lee
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 2.1
  • Lemma 2.2: McDiarmid's inequality
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 5 more