Bunkbed conjecture

The Bunkbed Conjecture (also spelled Bunk Bed Conjecture) is a statement in percolation theory, a branch of mathematics that studies the behavior of connected clusters in a random graph. The conjecture is named after its analogy to a bunk bed structure. It was first posited by Kasteleyn.

Description
The conjecture has many equivalent formulations. In the most general formulation it involves two identical graphs, referred to as the 'upper bunk' and the 'lower bunk'. These graphs are isomorphic, meaning they share the same structure. Additional edges, termed 'posts', are added to connect each vertex in the upper bunk with the corresponding vertex in the lower bunk.

Each edge in the graph is assigned a probability. The edges in the upper bunk and their corresponding edges in the lower bunk share the same probability. The probabilities assigned to the posts can be arbitrary.

A random subgraph of the bunkbed graph is then formed by independently deleting each edge based on the assigned probability.

Statement of the conjecture
The Bunkbed Conjecture states that in the resulting random subgraph, the probability that a vertex $$x$$ in the upper bunk is connected to some vertex $$y$$ in the upper bunk is greater than or equal to the probability that $$x$$ is connected to $$z$$, the isomorphic copy of $$y$$ in the lower bunk.

Interpretation and significance
The conjecture suggests that two vertices of a graph are more likely to remain connected after randomly removing some edges if the graph distance between the vertices is smaller. This is intuitive, but proving this conjecture is not straightforward and is an active area of research in percolation theory. Recently, it was resolved for particular types of graphs, such as wheels, complete graphs, complete bipartite graphs and graphs with a local symmetry. It was also proven in the limit $$p \to 1$$ for any graph