241 problems

Let $P$ be a polytope, with a vertex/edge graph whose diameter is measured in graph distance. The Polynomial Hirsch Conjecture is a central problem of Discrete Geometry, stating th…

Mihail–Vazirani conjecture. Every $0/1$ polytope $P$ satisfies

Ceballos–Pons geometric realization conjecture. The Hasse diagram of the ${\boldsymbol{s}}$-weak order can be realized as an orientation of the skeleton of a polyhedral subdivision…

Klee's conjecture. The upper bound conjecture holds for all Eulerian complexes.

Let $\Delta_{d-1}$ be the standard $(d-1)$-simplex, and let $n\Delta_{d-1}$ denote its dilation by $n$. A unit simplex is a simplex of normalized volume one in $n\Delta_{d-1}$, and…

Let $\mathcal{L}\in\mathcal{D}^k$ be a family of $k$ matroids on $V$, and let $w\in\mathbb{R}_{\ge0}^V$. Weighted fractional matching-cover conjecture. One has … This is presented…

Bohn–Faenza–Fiorini–Fisikopoulos–Macchia–Pashkovich conjecture. The cube and cross-polytope are the only 2-level polytopes for which equality is attained. The supplied text does no…

Let $S\in\{0,1\}^{m\times n}$. A matrix is a slack matrix if it is the slack matrix of a polytope. Polynomial-time recognition conjecture. There is an algorithm polynomial in $m,n$…

Let $d=3$ and $n=2k$, and let $\widehat{\Sigma}_k$ denote the $k$th higher secondary polytope of the configuration. Farber's conjecture. The diameter of $\widehat{\Sigma}_k$ is … T…

Ardila–Develin's correspondence conjecture. Tropical oriented matroids are in bijection with subdivisions of

Vertex degeneration conjecture. For any vertex $v$ in the generic marked poset polytope, there exists a vertex $t_v$ of the hypercube $[0,1]^{\tilde P}$ such that the image of $v$…

Grötschel–Padberg conjecture. For every integer $n\geq 5$, the diameter of the skeleton of the symmetric traveling salesperson polytope is

Non-revisiting Conjecture. There is a path from $u$ to $v$ which at every step enters a different facet of $P$.

Let $P$ be a $d$-dimensional $0/1$-polytope, meaning the convex hull of a subset of the vertices of the $d$-cube. Facet-count conjecture. The polytope $P$ cannot have more than … f…

3-geodesic conjecture. For every $d$-dimensional Dantzig figure $(P,x,y)$,

Strong Dantzig conjecture. Every fundamental deformation of a $d$-dimensional Dantzig figure is good; equivalently,

Dantzig conjecture. The graph $E(d)$ is strongly connected: for any vertices $x,y$ of $E(d)$, there is an oriented path from $x$ to $y$.

Face-structure conjectures for tropical polytopes. The following assertions hold: (1) $k$-faces of tropical polytopes are extreme sets; (2) the topological boundary of a $k$-face i…

Tropical halfspace characterization conjecture. A tropical polytope $P$ is pure and full dimensional if and only if it has a halfspace description whose apices are in general posit…

Tropical halfspace representation conjecture. If $P$ is pure, then the halfspaces from a generic lift $\overline{P}$ of $P$ map to tropical halfspaces whose intersection is $P$ its…

Combinatorial characterization conjecture. If the assignment $M\mapsto G_M$ satisfies 1. for every such decomposition,

The f-vector conjecture. The subdivision $\mathcal{D}_P$ has at most

UBC for the facets of geometric simplex triangulations. Every geometric triangulation $\mathcal S$ of $\mathbf{s}$ with vertex set $\mathcal V$ has at most

Let $\mathrm{met}_n$ be the metric polytope, and let its fractional vertices be the vertices that are not integral. The restriction of $\mathrm{met}_n$ to its fractional vertices i…

Let $G=(V,E)$ be a 2-colored graph with coloring ${\mathcal C}:E\to\{R,B\}$, and let $y\in\mathbb{N}^E$. The vector $y$ is a nonnegative integral combination of characteristic vect…