9 problems

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…

Reduced-expression isometric-dimension conjecture.

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.

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.