7 problems

Let $G$ be a graph in the class $\mathscr{H}$ of graphs whose cut semigroup has the Hilbert basis property, and let $f\in E(G)$ be an edge. Write $G\setminus f$ for deletion of $f$…

Exactness conjecture. The second semidefinite lifting is exact for $\mathrm{CUT}^4_\infty$, i.e.,

Let $G(V,E)$ be a graph, and let $\operatorname{Cut}^\square(G)$ be the convex hull of the cut points $\delta_{A\vert B}\in\mathbb{R}^{\lvert E\rvert}$ over all unordered vertex pa…

Normality–$K_5$-minor conjecture. The following conditions are equivalent:

The grid-graph conjecture. The cut polytope ${\rm Cut}^\square(G)$ is normal if $G$ is a grid graph.

The 4-connected plane triangulation conjecture. The cut polytope ${\rm Cut}^\square(G)$ is normal if $G$ is a 4-connected plane triangulation.

Sturmfels–Sullivant's conjecture. The cut polytope ${\rm Cut}^\square(G)$ is normal if and only if $G$ has no $K_5$ minor.