9 problems

Equivistal subdivision conjecture. The equivalence relation induced by equivistality constitutes a convex polyhedral subdivision of $S$. Moreover, the number of open regions in thi…

Polynomial combinatorial-type conjecture. The cardinality of the set of combinatorial types of shortest paths in $S$ is polynomial in the number of facets of $S$ when the dimension…

Let $G$ be a plane graph, and let $P$ be a set of non-crossing single-touch shortest paths in $G$. The path covering with forests number of $P$, denoted by $\operatorname{PCFN}(P)$…

Converse compatibility conjecture. If $\Sigma_1$ and $\Sigma_2$ are two compatible signed graphs, then the connected tensor product $\Sigma_1\otimes\Sigma_2$ is compatible.

Let $p\geq 3$, let $G^{(p)}_{n+1}$ contain the copy $AG^{(p)}_n$, and let $S_{n_0}$ be the recursively defined border set. Let $a$ be the vertex labeled $A^{n_0+1}$, and call a ver…

Let $p\geq 3$, let ${\cal A}_p$ be an alphabet of $p$ letters, and let $A,B\in{\cal A}_p$. For $n\geq 2$, consider vertices labeled $A^n$ and $B^n$. A vertex on a path is special i…

Let $G(n,\pi)$ be the unit disk graph, and let two vertices be displaced by $r_{xy}$. Write $\sigma_{r_{xy}}$ for the number of geodesic paths between them. Negative-binomial conje…

Let $G$ be a large but finite graph with negative curvature. Let the demand and inertia of a vertex be the graph quantities defined in the paper. Jonckheere–Lou–Bonahon–Baryshnikov…

Let $G$ be a large but finite graph with negative curvature. Let the demand of a vertex mean the quantity measuring how many shortest paths pass through it, as defined in the paper…