15 problems

Isometric-dimension conjecture. If $(W,S)$ is type $\Lambda$ and ${\boldsymbol{\alpha}}$ is a reduced expression, then

Fukuda–Handa conjecture. Every harmonic-even partial cube is antipodal.

Let $G$ be a partial cube, and let $c(G)$ denote its cop number. Suppose that $G$ isometrically embeds into the $n$-dimensional hypercube $Q_n$. Partial-cube cop-number conjecture.…

Let $G$ be a vertex-transitive partial cube with girth $6$. Marc's conjecture. Doubled Odd graphs are the only vertex-transitive partial cubes with girth $6$. This conjecture propo…

Simply-laced partial-cube conjecture. The conclusion of Theorem holds for all simply-laced Coxeter systems, without requiring the Coxeter graph to be triangle-free.

Djoković–Winkler classes conjecture. The equivalence classes of the Djoković–Winkler relation are precisely the collections of edges corresponding to specific braid shadows.

Let a complex of oriented matroids (COM) be a set family equipped with its VC-dimension, and let an ample partial cube be a partial cube with the ample property. A completion of a…

Let a complex of oriented matroids (COM) be a set family of VC-dimension $d$, and let an ample completion be an ample partial cube containing it. COM ample-completion conjecture. A…

Let $G$ be a Mandel oriented matroid of rank $r$, and let $E$ be any $\Theta$-class of its tope graph. Theta-class strengthening of Murty's conjecture. The $\Theta$-class $E$ is in…

Let $G$ be the tope graph of an oriented matroid (OM), and let $r(G)$ be its rank. A vertex $v$ is simplicial if $\deg(v)=r(G)$. Las Vergnas's conjecture. Every OM has a simplicial…

Gated-hull characterization conjecture. If the gated hull of each isometric cycle of $G$ is a disk or a full subdivision, then $G$ is two-dimensional.

Retraction characterization of hyper-median partial cubes. A partial cube is hyper-median if and only if it is a retract of a Cartesian product of bipartite cellular graphs.

Convex-extension conjecture. Every $G\in\mathcal{G}_{\rm COM}$ is a convex subgraph of a graph $G'\in\mathcal{G}_{\rm OM}$.

The retract characterization conjecture. A partial cube $G$ is hypercellular if and only if $G$ is a retract of a Cartesian product of bipartite cellular graphs.

Middle-level graph conjecture. The graph $G$ is a middle-level graph.