21 problems

Let $A$ be a dense $n$-element point set in general position in the plane, and let $K_n[A]$ denote the complete geometric graph induced by $A$. A path is monotone if its edge set i…

Let a finite planar point configuration be a finite set of points in the plane, where the closest neighbors of a point are the points minimizing Euclidean distance from it. Karabeg…

Let $k$ be a positive integer, and let $G$ be a complete geometric graph on $2k$ points. A $k$-staircase is the point-set construction defined earlier in the source. Staircase char…

Let $n\ge3$ be odd, and let $G$ be a complete geometric graph on $n$ vertices that can be decomposed into $\left\lceil\frac{n}{2}\right\rceil+1$ plane star-forests. Convex-hull-siz…

A complete geometric graph is a complete graph drawn with vertices in general position and straight-line edges. A plane star-forest is a forest whose connected components are stars…

A complete geometric graph is a complete graph drawn with vertices in general position and straight-line edges. A plane $k$-star-forest is a star-forest with at most $k$ connected…

Plane star-forest covering conjecture. There is no complete geometric graph with $n$ vertices that can be decomposed into fewer than

Let $S$ be a point set with $3n$ elements in the plane in general position. An intersecting edge-disjoint triangle family is a family of edge-disjoint triangles such that every two…

Let $S$ be a point set in the plane in general position, and let a longest Hamiltonian cycle be a Hamiltonian cycle on $S$ whose total edge length is maximal. Longest-cycle crossin…

Let $S$ be a set of $n$ points in the plane in general position. An elbow is an orthogonal geometric-graph edge consisting of one horizontal and one vertical line segment, and a cr…

Let $G$ be a geometric graph with $2|E(G)|\geq |V(G)|$. The disjoint-edge lower-bound conjecture. Then … This conjecture gives a lower bound for the number of disjoint edge pairs i…

Let $m$ be a non-negative integer and $G$ be a geometric graph such that $|DJ(uv)|\leq m$ for any edge $uv\in E(G)$. The disjoint-edge bound. Then … The conjecture proposes a sharp…

A grounded segment graph is the intersection graph of a collection of line segments grounded on a common line. Grounded segment-graph conjecture. The class of grounded segment grap…

A finite set $P\subset\mathbb{R}^d$ defines a $k$-nearest-neighbour graph by joining two points whenever one is among the $k$ points closest to the other. Here,…

Let $G$ be a unit disk graph, let $\zeta(G)$ denote its localization number, and let $\omega(G)$ denote its clique number. Unit-disk localization conjecture. There is a function…

Let $\mathsf{K}_n$ be a complete geometric graph and let $[\mathsf{K}_n,P]$ be a decomposition. Disjoint-triangles conjecture. If $P$ contains at least … triangles, then $P$ contai…

Let $\mathsf{K}_n$ be a complete geometric graph of order $n$, and let $[\mathsf{K}_n,P]$ be a decomposition. The clique index $\omega'([\mathsf{K}_n,P])$ is the maximum size of a…

A geometric graph is a graph drawn in the plane with vertices represented by points and edges as straight-line segments. A circle graph $F$ is the intersection graph of chords on a…

A geometric graph is a graph drawn in the plane with vertices represented by points and edges as straight-line segments. For integers $k,l\geq 1$, a $(k,l)$-crossing family is a pa…

Let $t(n)$ be the minimum number of colours needed to colour the edges of some geometric drawing of $K_n$ so that every pair of edges with the same colour crosses; each colour clas…

$\frac{3\pi}{2}$-open spanning path conjecture. Every finite point set in general position in the plane has a $\frac{3\pi}{2}$-open spanning path.