相关论文: Transversal structures on triangulations: a combin…

Given a combinatorial triangulation of an $n$-gon, we study (a) the space of all possible drawings in the plane such the edges are straight line segments and the boundary has a fixed shape, and (b) the algebraic variety of possibilities for…

A plane graph is said to be a rectangular graph if each of its edges can be oriented horizontal or vertical, its internal regions are four-sided and it has a rectangular enclosure. If dual of a planar graph is a rectangular graph, then the…

Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via…

Rectangulations are partitions of a square into axis-aligned rectangles. A number of results provide bijections between combinatorial equivalence classes of rectangulations and families of pattern-avoiding permutations. Other results deal…

We propose a simple geometrical construction of topological invariants of 3-strand Brownian braids viewed as world lines of 3 particles performing independent Brownian motions in the complex plane z. Our construction is based on the…

Convexification techniques have gained increasing interest over the past decades. In this work, we apply a recently developed convexification technique for fractional programs by He, Liu and Tawarmalani (2024) to the problem of determining…

We introduce and study a new graph representation where vertices are embedded in three or more dimensions, and in which the edges are drawn on the projections onto the axis-parallel planes. We show that the complete graph on $n$ vertices…

Motivated by his work on the classification of countable homogeneous oriented graphs, Cherlin asked about the typical structure of oriented graphs (i) without a transitive triangle, or (ii) without an oriented triangle. We give an answer to…

Stripe patterns are ubiquitous in nature and everyday life. While the synthesis of these patterns has been thoroughly studied in the literature, their potential to control the mechanics of structured materials remains largely unexplored. In…

Inspired by a width invariant on permutations defined by Guillemot and Marx, Bonnet, Kim, Thomass\'e, and Watrigant introduced the twin-width of graphs, which is a parameter describing its structural complexity. This invariant has been…

The main purpose of this paper is twofold. We first want to analyze in details the meaningful geometric aspect of the method introduced in the previous paper [12], concerning regularity of families of irreducible, nodal "curves" on a…

We give a parity reversing involution on noncrossing trees that leads to a combinatorial interpretation of a formula on noncrossing trees and symmetric ternary trees in answer to a problem proposed by Hough. We use the representation of…

In the paper, we consider the rigidity problem of the infinite hexagonal triangulation of the plane under the piecewise linear conformal changes introduced by Luo in [5]. Our result shows that if a geometric hexagonal triangulation of the…

Recently, Ian Agol introduced a class of "veering" ideal triangulations for mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the singular points. These triangulations have very special combinatorial properties, and…

Flips in triangulations have received a lot of attention over the past decades. However, the problem of tracking where particular edges go during the flipping process has not been addressed. We examine this question by attaching unique…

Many interesting geometric structures can be described as regular infinitesimal flag structures, which occur as the underlying structures of parabolic geometries. Among these structures we have for instance conformal structures, contact…

It is well known that a triangle with side lengths 3, 4 and 5 is right-angled. Euclid was the first to give a formula for generating other right-angled triangles with integer side lengths. In this text, I present a novel algorithm to…

Via circle pattern techniques, random planar triangulations (with angle variables) are mapped onto Delaunay triangulations in the complex plane. The uniform measure on triangulations is mapped onto a conformally invariant spatial point…

A pseudo-triangle is a simple polygon with three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as…

Inspired by the seminal result that a graph and an associated rotation system uniquely determine the topology of a closed manifold, we propose a combinatorial method for reconstruction of surfaces from points. Our method constructs a…