8 problems

A plane triangulation is a loopless plane graph in which every face has size three. Let $P(G,t)$ be its chromatic polynomial, let $\tau=(1+\sqrt{5})/2$ be the golden ratio, and let…

Let $T$ be a plane triangulation of minimum degree at least $6$, let $\mathbb T_6$ be the Euclidean triangular lattice, let $p_{\mathrm c}(T)$ be the critical probability for bond…

Let $T$ be a plane triangulation of minimum degree at least $6$, let $\mathbb T_6$ be the Euclidean triangular lattice, and let $\mu(T)$ denote the exponential growth rate of the n…

Shedding-diameter conjecture. With high probability,

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…

Tetrahedral embedding conjecture. Each graph $G\in{\cal P}$ with one point orbit can be embedded on the sphere of the solid regular tetrahedron in such a way that all four induced…

A triangulation is a plane graph in which every face, including the outer face, is bounded by a triangle. The degree of a vertex is its number of incident edges. The bounded-except…

A triangulation is a plane graph in which every face, including the outer face, is bounded by a triangle. A dominating set of a graph is a set containing every vertex or a neighbor…