Disjoint faces in simple drawings of the complete graph and topological Heilbronn problems

12/02/2022
by   Alfredo Hubard, et al.
0

Given a complete simple topological graph G, a k-face generated by G is the open bounded region enclosed by the edges of a non-self-intersecting k-cycle in G. Interestingly, there are complete simple topological graphs with the property that every odd face it generates contains the origin. In this paper, we show that every complete n-vertex simple topological graph generates at least Ω(n^1/3) pairwise disjoint 4-faces. As an immediate corollary, every complete simple topological graph on n vertices drawn in the unit square generates a 4-face with area at most O(n^-1/3). Finally, we investigate a ℤ_2 variant of Heilbronn triangle problem.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
07/16/2023

Short edges and noncrossing paths in complete topological graphs

Let h(n) be the minimum integer such that every complete n-vertex simple...
research
08/31/2019

Simple k-Planar Graphs are Simple (k+1)-Quasiplanar

A simple topological graph is k-quasiplanar (k≥ 2) if it contains no k p...
research
01/06/2020

(Theta, triangle)-free and (even hole, K_4)-free graphs. Part 2 : bounds on treewidth

A theta is a graph made of three internally vertex-disjoint chordless p...
research
11/21/2019

Implementing the Topological Model Succinctly

We show that the topological model, a semantically rich standard to repr...
research
04/08/2022

List covering of regular multigraphs

A graph covering projection, also known as a locally bijective homomorph...
research
12/28/2018

The complete set of minimal simple graphs that support unsatisfiable 2-CNFs

A propositional logic sentence in conjunctive normal form that has claus...
research
08/31/2020

On Polyhedral Realization with Isosceles Triangles

Answering a question posed by Joseph Malkevitch, we prove that there exi...

Please sign up or login with your details

Forgot password? Click here to reset