13 problems

Product-structure conjecture for $k$-independent crossing graphs. The class of $k$-independent crossing graphs admits product structure.

The no-product-structure conjecture. For every $\alpha < s(n)$, the class of intersection graphs of $\alpha$-free homothetic regular $n$-gons does not have product structure.

The canonical-drawing characterization. The class of intersection graphs of $\alpha$-free homothetic regular $n$-gons admits product structure if and only if their canonical drawin…

Uncrossed-number separation conjecture. The uncrossed number can be arbitrarily far apart from the outerthickness. This conjecture asks whether the difference between these two gra…

Rectangular-flat-torus grid contact conjecture. Every bipartite toroidal graph without loops has a grid contact representation on the rectangular flat torus.

Rectangular-flat-torus tessellation conjecture. Every toroidal graph without loops has a tessellation representation on the rectangular flat torus.

Potential crossing pair conjecture. The graph $G$ has crossing number at least $2$ if and only if it does not have a potential crossing pair.

Let $X_n$ denote the gyroelongated $n$-bipyramid, and let ${\operatorname{{obs}}}_\mathrm{out}(X_n)$ be its outside obstacle number. Obstacle-number conjecture. If $n\geq 4$, then…

Crossing-number equality conjecture.

Rectilinear crossing-number conjecture.

Let $D$ be a simple drawing of a graph, and let $E_{{\le}{\le}{\le}k}(D)$ denote the number of its $\le\!\le\!\le k$-edges, with … Here $E_i(D)$ denotes the number of $i$-edges in…

Let $D$ be a simple drawing of a graph, and let $E_{{\le}{\le}{\le}k}(D)$ denote the number of its $\le\!\le\!\le k$-edges, defined by … Here $E_i(D)$ denotes the number of $i$-edg…

A drawing of a graph represents vertices by distinct plane points and edges by simple curves, with a vertex meeting an edge only at an endpoint. A drawing is simple if any two edge…