11 problems

Let $f(w)$ be the spanning-forest free energy per site on a regular lattice of dimension $d\geq2$ and coordination number $r$, and let $f_0$ denote the entropy per site for spannin…

Uniform forest p-NC conjecture. The measure $\mathbb{P}_{\mathrm{UF}}$ satisfies the p-NC property.

Mohr–Pardey–Rautenbach conjecture. There exists a copy $F'$ of $F$ in $K_n$ such that

Let $m$ be the number of marked vertices in a graph. A quadratic spanning forest identity of the indicated type has coefficients as free variables, where the left-hand side consist…

Let $G$ be a graph, and let $\mathbf{F}$ be a uniformly random forest chosen from all forests of $G$. For edges $e,f\in E(G)$, negative correlation conjecture. … This conjecture as…

Let $G$ be a $d$--regular graph on $n$ vertices, and let $F(G)$ denote its number of spanning forests. Bencs–Csikvári conjecture. … This is the proposed sharp analogue for spanning…

Let $K_n$ be a complete graph of order $n$, and let $c:E(K_n)\to \{-1,1\}$ be a zero-sum labeling satisfying … Let $F$ be a spanning forest of $K_n$ with maximum degree $\Delta$. M…

Let $K_n$ be a complete graph, let $c:E(K_n)\to\{-1,1\}$ be a zero-sum labeling satisfying … and let $F$ be a spanning forest of $K_n$. Write $\Delta(F)$ for the maximum degree of…

Let $\mathcal G$ be a graph. Say that $\mathcal G$ is $\Delta$-SOS if every relevant Rayleigh difference $\Delta G\{e,f\}$ has the required sum-of-squares representation, and say t…

Let $\mathcal G$ be a graph, and let $e$ and $f$ be distinct edges. Write $G=F(\mathcal G;\mathbf y)$ for the generating polynomial of its spanning forests, and let $\mathscr S$ be…

Let $G$ be a (weighted) graph. Let $\mathcal{F}$ be the set of all spanning forests of $G$, and let $\mathcal{B}$ be the probability distribution on spanning forests in which each…