130 problems

Hamiltonicity conjecture. The graph $G(L(P))$ has a Hamiltonian cycle.

Let $\mathcal{G}$ be a directed acyclic graph with skeleton $G$, and let $i$ be a vertex. Define $\mathcal{G}_{\downarrow i}$ by orienting every edge between $i$ and a neighbor tow…

Let $\mathsf{PF}_n$ denote the parking function polytope. A triangulation of a lattice polytope is regular if it is induced by a lifting function, and unimodular if every simplex i…

Let $m\in\mathbb{P}$, and let $c(\operatorname{\mathcal{W}}(B_{2m}))$ be the convex hull of the signed Wachs permutations in $\mathbb{R}^{2m}$. Simplicity conjecture. The polytope…

Let $m\in\mathbb{P}$, and let $c(\operatorname{\mathcal{W}}(S_{2m}))$ be the convex hull of the Wachs permutations in $\mathbb{R}^{2m}$. Simplicity conjecture. The polytope…

NP-hardness conjecture. It is NP-hard to decide if a given $N$-bit vector $f=(1,f_0,\ldots,f_{d-1})$ of positive integers is the $f$-vector of a $d$-polytope.

Enumeration conjecture for ${\cal A}_n^j$. The number of elements of ${\cal A}_n^j$ is

Chan–Robbins–Yuen volume conjecture. The relative volume of $P_n$ is

Kalai–Kleinschmidt–Lee conjecture. For every $j$, the number of $j$-dimensional empty simplices of $P$ is at most the number of $j$-dimensional empty simplices of…

Let an $m$-prism be triangulated using $t$ corner tetrahedra. Corner-tetrahedra conjecture. Such a triangulation has at least … tetrahedra, and therefore at least … interior triang…

Let $R$ be a commutative ring and let $P$ be a Col-divisible polytope of arbitrary dimension. Write $K_i(R,P)$ for its associated $K$-groups, and let ${\mathfrak c}(P)\leq\dim P$ b…

Let $W$ be a finite Coxeter group and let $\stackrel{\to}{G}$ be an orientation of its Coxeter diagram. The Cambrian fan … is the fan whose maximal cones are the unions of reflecti…

Let $T$ be a rooted tree, let ${\mathcal A}_T$ be the configuration of exponent vectors from the toric parametrization, and let $P_T$ be the convex hull of the columns of…

Spanning conjecture. The set of $f$-vectors of all ordinary $d$-polytopes spans the Euler hyperplane. A spanning set consists of the ordinary polytopes

Let $\Delta\subset\mathbb{R}^d$ be a polytope, let $\Phi$ be its dual fan, and let $\overline{M}_\Phi$ be the quotient combinatorial intersection cohomology module defined from the…

Let $\Phi$ be a complete fan in a vector space $V$ of dimension $n$. Let $\Omega^1_\Phi$ be the sheaf with stalk $\Omega^1_{\Phi,\sigma}=\operatorname{Span}(\sigma)^\perp$, and let…

Let $P$ be a primitive polytope in $\mathbb{E}^d$, where $d\geq 4$. Primitive-polytope vertex conjecture. The polytope $P$ has at most $2^d$ vertices, and it has fewer than $2^d$ v…

Let $K\in\mathcal{K}^d$ be a convex body, and say that a set of directions illuminates $K$ if every point of $\operatorname{bd}K$ is illuminated by at least one direction in the se…

For a $d$-polytope, let the transversal ratios be denoted by $\rho_d^\mathrm{P}$ and $\rho_d^\mathrm{S}$, corresponding to polytopes and simplicial polytopes, respectively. A polyt…

Let $n \geq 5$ be such that $n \not\equiv 2 \pmod 4$, and set $k = \floor*{n/2}$. The polytope $P_\mathrm{QTSASM}$ is the polytope of quarter-turn symmetric alternating sign matric…

Let $d\geqslant 3$ and $1\leqslant\ell\leqslant d-1$. Let $P$ be a $d$-polytope with $2d+\ell$ vertices, and let $T(r)$ denote the $r$-simplex. Pineda-Villavicencio's conjecture. 1…

Let $P$ be a simple polytope, and let $(\gamma_i(P))$ and $(g_i(P))$ denote its $\gamma$-vector entries and toric $g$-vector entries, respectively. Toric g-vector Kruskal–Katona co…

Central-slice extremality conjecture. For every centrally symmetric $k$-dimensional polytope $P$, the upper bound on the number of vertices of a $(k-1)$-dimensional slice of $P$ is…

Generic central slice conjecture. The upper bound on the number of vertices of a $k$-dimensional slice of $C_d$ can be attained by a generic central $k$-dimensional slice for every…

Circuit Distance conjecture. The Circuit Distance problem is NP-hard for polygons.