Uli Wagner
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Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs Open
A linearly ordered (LO) k -colouring of a hypergraph is a colouring of its vertices with colours 1, …, k such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO k -colouring with a fixed number …
Hardness of 4-Colouring 𝐺-Colourable Graphs Open
We study the complexity of a class of promise graph homomorphism problems. For a fixed graph H, the H-colouring problem is to decide whether a given graph has a homomorphism to H. By a result of Hell and Nešetřil, this problem is NP-hard f…
Eight-Partitioning Points in 3D, and Efficiently Too Open
An eight-partition of a finite set of points (respectively, of a continuous mass distribution) in $$\mathbb {R}^3$$ consists of three planes that divide the space into 8 octants, such that each open octant contains at most 1/8 of the point…
Eight-Partitioning Points in 3D, and Efficiently Too Open
An eight-partition of a finite set of points (respectively, of a continuous mass distribution) in ℝ³ consists of three planes that divide the space into 8 octants, such that each open octant contains at most 1/8 of the points (respectively…
Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs Open
A linearly ordered (LO) k-colouring of a hypergraph is a colouring of its vertices with colours 1, … , k such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO k-colouring with a fixed number o…
Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes Open
We prove the following quantitative Borsuk–Ulam-type result (an equivariant analogue of Gromov’s Topological Overlap Theorem): Let X be a free ℤ/2-complex of dimension d with coboundary expansion at least η k in dimension 0 ≤ k 0 of pairs …
Eliminating Higher-Multiplicity Intersections, III. Codimension 2 Open
We study conditions under which a finite simplicial complex $K$ can be mapped to $\mathbb R^d$ without higher-multiplicity intersections. An almost $r$-embedding is a map $f: K\to \mathbb R^d$ such that the images of any $r$ pairwise disjo…
Connectivity of Triangulation Flip Graphs in the Plane Open
Given a finite point set P in general position in the plane, a full triangulation is a maximal straight-line embedded plane graph on P. A partial triangulation is a full triangulation of some subset P' of P containing all extreme points in…
Connectivity of Triangulation Flip Graphs in the Plane (Part II: Bistellar Flips) Open
Given a finite point set P in general position in the plane, a full triangulation is a maximal straight-line embedded plane graph on P. A partial triangulation on P is a full triangulation of some subset P' of P containing all extreme poin…
Shellability is NP-complete Open
We prove that for every d ≥ 2, deciding if a pure, d -dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every …
The Crossing Tverberg Theorem Open
Tverberg's theorem is one of the cornerstones of discrete geometry. It states that, given a set $X$ of at least $(d+1)(r-1)+1$ points in $\mathbb R^d$, one can find a partition $X=X_1\cup \ldots \cup X_r$ of $X$, such that the convex hulls…
Computing simplicial representatives of homotopy group elements Open
A central problem of algebraic topology is to understand the homotopy groups of a topological space X. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group of a g…
On the Treewidth of Triangulated 3-Manifolds Open
In graph theory, as well as in 3-manifold topology, there exist several width-type parameters to describe how "simple" or "thin" a given graph or 3-manifold is. These parameters, such as pathwidth or treewidth for graphs, or the concept of…
A Proof of the Orbit Conjecture for Flipping Edge-Labelled\n Triangulations Open
Given a triangulation of a point set in the plane, a \\emph{flip} deletes an\nedge $e$ whose removal leaves a convex quadrilateral, and replaces $e$ by the\nopposite diagonal of the quadrilateral. It is well known that any triangulation\no…
On Generalized Heawood Inequalities for Manifolds: a van Kampen--Flores-type Nonembeddability Result Open
The fact that the complete graph $K_5$ does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete grap…
Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the $r$-Metastable Range Open
Motivated by Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without higher-mult…
Eliminating higher-multiplicity intersections, II. The deleted product criterion in the r-metastable range Open
Motivated by Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without higher-mult…