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Bunkbed conjecture for complete bipartite graphs and related classes of graphs

Thomas Richthammer

Abstract

Let $G = (V,E)$ be a simple finite graph. The corresponding bunkbed graph $G^\pm$ consists of two copies $G^+ = (V^+,E^+),G^- = (V^-,E^-)$ of $G$ and additional edges connecting any two vertices $v_+ \in V_+,v_- \in V_-$ that are the copies of a vertex $v \in V$. The bunkbed conjecture states that for independent bond percolation on $G^\pm$, for all $v,w \in V$, it is more likely for $v_-,w_-$ to be connected than for $v_-,w_+$ to be connected. While this seems very plausible, so far surprisingly little is known rigorously. Recently the conjecture has been proved for complete graphs. Here we give a proof for complete bipartite graphs, complete graphs minus the edges of a complete subgraph, and symmetric complete $k$-partite graphs.

Bunkbed conjecture for complete bipartite graphs and related classes of graphs

Abstract

Let be a simple finite graph. The corresponding bunkbed graph consists of two copies of and additional edges connecting any two vertices that are the copies of a vertex . The bunkbed conjecture states that for independent bond percolation on , for all , it is more likely for to be connected than for to be connected. While this seems very plausible, so far surprisingly little is known rigorously. Recently the conjecture has been proved for complete graphs. Here we give a proof for complete bipartite graphs, complete graphs minus the edges of a complete subgraph, and symmetric complete -partite graphs.
Paper Structure (5 sections, 5 theorems, 25 equations)

This paper contains 5 sections, 5 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Vertices with the same neighbors: Theorem \ref{['Thm:new']}
  4. Special classes of graphs: Corollaries \ref{['Cor:bipartite']} and \ref{['Cor:kpartite']}
  5. Neighboring vertices: Theorem \ref{['Thm:old']}

Key Result

Theorem 1

The case of neighboring vertices with a local symmetry. Let $G = (V,E)$ be a finite complete graph with weights $p_G \in [0,1)^E \times [0,1]^V$. Let $v,w \in V$ such that $p_{vw} > 0$. In case of $p_v,p_w \neq 1$ suppose that for all $u \in V$ with $p_{uw} > 0$ and $p_{u} \neq 1$ there is an automo

Theorems & Definitions (9)

  • Definition 1
  • Remark 1
  • Theorem 1
  • Remark 2
  • Corollary 1
  • Theorem 2
  • Remark 3
  • Corollary 2
  • Corollary 3