34 problems

A thrackle is a simple topological graph in which every pair of edges that do not share a vertex crosses. Conway's thrackle conjecture. Every thrackle with $n$ vertices has at most…

Eulerian edge-refinement conjecture. $d$-spheres have edge refinements which are Eulerian.

Call a topological graph $k$-quasi-plane if it has no $k$ pairwise crossing edges. The quasi-plane graph linear density conjecture. For any integer $k \geq 2$ there is a constant…

Pach–Tóth conjecture. For every fixed $k$, the number of edges in such graphs is at most $O_k(n)$.

A topological graph $E$ has a vertex space $E^0$ and an associated $C^*$-algebra equipped with a natural circle-group gauge action. A tracial state on this $C^*$-algebra is gauge-i…

Kaufmann–Ueckerdt's density conjecture. The density of simple fan-planar topological graphs with non-homeomorphic parallel edges and non-trivial loops remains

Let $k\geq 4$ be even, and let $G$ be an $n$-vertex complete simple topological graph. A collection of $k$-faces is pairwise disjoint if no two faces in the collection intersect. T…

A complete $n$-vertex simple topological graph is a drawing of the complete graph on $n$ vertices in which vertices are points and edges are simple curves, with no edge passing thr…

Let $h(n)$ denote the maximum, over complete $n$-vertex simple topological graphs, of the minimum number of crossings of an edge in a suitable noncrossing matching construction. Sh…

Let $E$ be a free topological graph, and let $C^{*}(E)$ denote its graph $C^{*}$-algebra. A tracial state on $C^{*}(E)$ is a state $\tau:C^{*}(E)\to\mathbb{C}$ satisfying…

Let $\mathcal{P}$ be a set of $n$ points in the plane, and let there be $m$ topological trees such that every pair of trees intersects exactly once and every leaf of every tree bel…

Crossing-degree conjecture. There is an absolute constant $\varepsilon>0$ such that

Plane-path conjecture. There is an absolute constant $\varepsilon>0$ such that every complete $n$-vertex simple topological graph contains a plane path of length $n^{\varepsilon}$.

A separated multigraph is one in which no two parallel edges cross and the lens enclosed by every pair of parallel edges contains at least one vertex in its interior. A topological…

Let $T_{code}$ be a Topcode-matrix, with maximum edge-neighborhood size … A ve-proper non-negative integer Topcode-matrix has distinct endpoint labels in every column and distinct…

Let $T_{code}=(X~E~Y)^T$ be a Topcode-matrix, and let $(XY)^*$ denote its set of vertex labels. Define … A v-proper non-negative integer Topcode-matrix $T'_{code}=(X'~E'~Y')^T$ of…

Quasi-planar edge bound conjecture. One should have

Let $T$ be a lifted graph, let $F\in\mathcal{S}_1(T)$, and let $E$ be the finite exceptional set such that every rational number $p/q\in\operatorname{Rot}(F)\setminus E$ is realize…

Let $T$ be a lifted graph and let $F:T\to T$ be a map of degree one. Its rotation set is denoted by $\operatorname{Rot}(F)$. Whole rotation set conjecture. The whole rotation set…

Sphere coloring conjecture. Every $d$-sphere has chromatic number $d+1$ or $d+2$.

A simple complete topological graph is a drawing of a complete graph in the plane in which any two edges meet at most once, either at a common endpoint or at a proper crossing. A r…

Let $E$ be a free topological graph, and let ${\mathrm{C}^{*}}(E)$ denote its associated $C^*$-algebra. A tracial state on ${\mathrm{C}^{*}}(E)$ is a positive normalized linear fun…

A $k$-grid with distinct vertices in a topological graph is a pair of edge subsets $E_1,E_2$ such that $|E_1|=|E_2|=k$, every edge in $E_1$ crosses every edge in $E_2$, and the gri…

Let $e_4(n)$ denote the maximum number of edges in a topological graph with $n>2$ vertices in which every edge is involved in at most four crossings. Pach–Radoičić–Tardos–Tóth conj…

Unbounded-factor crossing lemma conjecture. The crossing points satisfy