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The bunkbed conjecture remains true when gluing along a vertex

Paul Meunier, Pegah Pournajafi

Abstract

We show that the bunkbed conjecture remains true when gluing along a vertex. As immediate corollaries, we obtain that the bunkbed conjecture is true for forests and that a minimal counterexample to the bunkbed conjecture is 2-connected.

The bunkbed conjecture remains true when gluing along a vertex

Abstract

We show that the bunkbed conjecture remains true when gluing along a vertex. As immediate corollaries, we obtain that the bunkbed conjecture is true for forests and that a minimal counterexample to the bunkbed conjecture is 2-connected.

Paper Structure

This paper contains 6 sections, 7 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Gluing along a vertex
  4. First case
  5. Second case
  6. Main theorem and corollaries

Key Result

Lemma 1

Let $\mu$ be a weight on $F$. Define the following weight $\mu'$ on $H$: for $e \in E(H)$, set Then, for every $x, y \in V(H)$, we have $\mathbb{P}_{F, \mu}(x \sim_F y) = \mathbb{P}_{H, \mu'}(x \sim_H y)$.

Figures (2)

  • Figure 1: The bunkbed graph of $G = C_4$.
  • Figure 2: The notation for Section 3.

Theorems & Definitions (15)

  • Conjecture 1: Bunkbed conjecture
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • ...and 5 more