12 problems

Let $\mathcal{H}$ be a hypergraph and let $T\subseteq V(\mathcal{H})$ be a set of posts. In the conditioned model $E_4^T(\widetilde{\mathcal{H}})$, precisely one of the two copies…

Let $G$ be a graph, and let $BB(G)$ be its bunkbed graph with two layers, whose corresponding vertices are denoted $x^-$ and $x^+$ for $x obreaktext{ in } obreak V(G)$. Let … be a…

Let $D$ be a simple digraph. Construct $\widetilde{D}$ by adding both directed vertical edges $v^{(0)}v^{(1)}$ and $v^{(1)}v^{(0)}$ for every $v\in V(D)$, and let $E_8$ select a su…

Let $D$ be a digraph, with the conditioned directed-edge model $E_6^{T,F}(\widetilde{D})$ and the corresponding unconditioned model $E_7(\widetilde{D})$ defined as in the source. L…

Let $Q$ be a graph, hypergraph, or digraph, and let $\mathcal{E}(\widetilde{Q})$ be a percolation model on its bunkbed structure. For vertices $u,v\in V(Q)$, write…

Let $G=(V,E)$) be a connected finite graph, let $L_2$ be the line graph on two vertices, and consider the bunkbed graph $G\square L_2$. For Bernoulli percolation with parameter…

Let a bunkbed graph have vertices $v^-$ and $v^+$ corresponding to each vertex $v$ of a base graph. Consider the continuous-time random walk on this bunkbed graph whose jump rate t…

Directed Bunkbed conjecture. For every $u,v\in V(G)$,

Hypergraph Bunkbed conjecture. For every $u,v\in V(G)$,

Two-color Bunkbed conjecture. For every $u,v\in V(G)$,

Generalized Bunkbed conjecture. For every $u,v\in V(G)$,

Bunkbed conjecture. For every $u,v\in V(G)$,