Related papers: Enumerating topological $(n_k)$-configurations

We construct a map from fundamental groups of complements to some plane configurations to the groups $G_{n}^{k}$ for large $k$. We discuss connection between the groups $G_{n}^{k}$ for different $G_{n}^{k}$ and their geometric realization.

Three types of geometric structure---grid triangulations, rectangular subdivisions, and orthogonal polyhedra---can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an…

Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…

We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n^{7/6}). No known input causes our algorithm to use area \Omega(n^{1+\epsilon}) for any…

For a graph $G$ and integer $k\geq1$, we define the token graph $F_k(G)$ to be the graph with vertex set all $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is a pair of adjacent…

A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for algebraic lines over some field or possibly real…

In this article we give a computational study of combinatorics of the discriminantal arrangements. The discriminantal arrangements are parametrized by two positive integers n and k such that n>k. The intersection lattice of the…

Beyond-planarity focuses on combinatorial properties of classes of non-planar graphs that allow for representations satisfying certain local geometric or topological constraints on their edge crossings. Beside the study of a specific graph…

Block designs are combinatorial structures in which each pair of a set of varieties appears together in a fixed number of blocks. Complete graphs are graphs in which every pair of vertices are adjacent. We present some new constructions of…

We construct tree-decompositions of graphs that distinguish all their k-blocks and tangles of order k, for any fixed integer k. We describe a family of algorithms to construct such decompositions, seeking to maximize their diversity subject…

Localization is a topological technique that allows us to make global equivariant computations in terms of local data at the fixed points. For example, we may compute a global integral by summing integrals at each of the fixed points. Or,…

In this paper we consider the generalized anchored configuration spaces on $n$ labeled points on a~graph. These are the spaces of all configurations of $n$ points on a~fixed graph $G$, subject to the condition that at least $q$ vertices in…

We organize a table of regular graphs with minimal diameters and minimal mean path lengths, large bisection widths and high degrees of symmetries, obtained by enumerations on supercomputers. These optimal graphs, many of which are newly…

To enumerate 3-manifold triangulations with a given property, one typically begins with a set of potential face pairing graphs (also known as dual 1-skeletons), and then attempts to flesh each graph out into full triangulations using an…

A continuous map from R^m to R^N or from C^m to C^N is called k-regular if the images of any $k$ points are linearly independent. Given integers m and k a problem going back to Chebyshev and Borsuk is to determine the minimal value of N for…

Matching is a method of the design of experiments. If we had an even number of patients and wanted to form pairs of patients such that their ages, for example, in each pair be as close as possible, we would use nonbipartite matching. Not…

We call a topological ordering of a weighted directed acyclic graph non-negative if the sum of weights on the vertices in any prefix of the ordering is non-negative. We investigate two processes for constructing non-negative topological…

We consider a simple game, the $k$-regular graph game, in which players take turns adding edges to an initially empty graph subject to the constraint that the degrees of vertices cannot exceed $k$. We show a sharp topological threshold for…

We study the ordered configuration spaces of star graphs. Inspired by the representation stability results of Church--Ellenberg--Farb for the ordered configuration space of a manifold and the edge stability results of…

In this paper we enumerate $k$-noncrossing tangled-diagrams. A tangled-diagram is a labeled graph whose vertices are $1,...,n$ have degree $\le 2$, and are arranged in increasing order in a horizontal line. Its arcs are drawn in the upper…