6 problems

Ellingham–Marshall–Ozeki–Royle–Tsuchiya conjecture. Let $G$ be a non-hamiltonian polyhedral graph with at least $16$ vertices. If $G$ has no $K_{2,6}$ minor, then $G$ is isomorphic…

A higher-rank tree is a higher-rank graph arising from the construction associated with a polyhedral graph, and it is planar when the underlying graph is embedded in a sphere. Assu…

Let $G=(V,E)$ be an involutive polyhedral graph: a self-dual polyhedral graph equipped with an involution $\tau$ satisfying $v\notin\tau(v)$ and $u\in\tau(v)\iff v\in\tau(u)$. A Re…

Smallest-graph conjecture. The graph $G_{RD}$ is the smallest non-regular, bipartite, polyhedral (3-connected), and transmission regular graph.

Let ${\mathcal{P}}_k^\text{irr}$ be the set of all irreducible polyhedral graphs with $k$ rigid vertices. Finiteness conjecture. Each set ${\mathcal{P}}_k^\text{irr}$ is finite. Th…

Polynomial grid-size conjecture. Every plane triangulation can be realized as the graph of a convex polyhedron embedded in an integer grid whose side lengths are bounded by a polyn…