Distance sets of well-distributed planar sets for polygonal norms

Optimal Polygonal Representation of Planar Graphs

In this paper, we consider the problem of representing graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also show that the lower bound of six sides is matched by an upper bound of six sides with a linear-time algorithm for representing any planar graph by touching hexagons. Moreover, our algorithm produces convex polygons with edges having at most three slopes and with all vertices lying on an O(n)xO(n) grid.

A simple symmetry generating operads related to rooted planar $m$-ary trees and polygonal numbers

The aim of this paper is to further explore an idea from J.-L. Loday briefly exposed in [5]. We impose a natural and simple symmetry on a unit action over the most general quadratic relation which can be written. This leads us to two families of binary, quadratic and regular operads whose free objects, as well as their duals in the sense of Ginzburg and Kapranov are computed. Roughly speaking, free objects found here are in relation to $m$-ary trees, triangular numbers and more generally $m$-tetrahedral numbers, homogeneous polynomials on $m$ commutative indeterminates over a field $K$ and pol…

覆盖每个平面方形网格的最短多边形链

Shortest polygonal chains covering each planar square grid

本文构造性地证明了平面方形网格存在与最小链覆盖路径链长相同的覆盖路径和回路，提出通用算法生成偶数网格的覆盖回路，并给出最小总行程距离的紧上界。

Obstacle Numbers of Planar Graphs

Given finitely many connected polygonal obstacles $O_1,\dots,O_k$ in the plane and a set $P$ of points in general position and not in any obstacle, the {\em visibility graph} of $P$ with obstacles $O_1,\dots,O_k$ is the (geometric) graph with vertex set $P$, where two vertices are adjacent if the straight line segment joining them intersects no obstacle. The obstacle number of a graph $G$ is the smallest integer $k$ such that $G$ is the visibility graph of a set of points with $k$ obstacles. If $G$ is planar, we define the planar obstacle number of $G$ by further requiring that the visibility …

On the number of lattice convex chains

An asymptotic formula is presented for the number of planar lattice convex polygonal lines joining the origin to a distant point of the diagonal. The formula involves the non-trivial zeros of the zeta function and leads to a necessary and sufficient condition for the Riemann Hypothesis to hold.

Lattice convex chains in the plane

A detailed combinatorial analysis of planar lattice convex polygonal lines is presented. This makes it possible to answer an open question of Vershik regarding the existence of a limit shape when the number of vertices is constrained. The method which is used emphasizes the connection of the combinatorial analysis with the zeros of the zeta function. It is shown how the Riemann Hypothesis leads to an asymptotic equivalent of the number of convex chains.