Empty Triangles in Good Drawings of the Complete Graph

A good drawing of a simple graph is a drawing on the sphere or, equivalently, in the plane in which vertices are drawn as distinct points, edges are drawn as Jordan arcs connecting their end vertices, and any pair of edges intersects at most once. In any good drawing, the edges of three pairwise con...

Full description

Saved in:
Bibliographic Details
Published in:Graphs and combinatorics Vol. 31; no. 2; pp. 335 - 345
Main Authors: Aichholzer, Oswin, Hackl, Thomas, Pilz, Alexander, Ramos, Pedro, Sacristán, Vera, Vogtenhuber, Birgit
Format: Journal Article Publication
Language:English
Published: Tokyo Springer Japan 01.03.2015
Springer Nature B.V
Subjects:
05 Combinatorics
05B Designs and configurations
Classificació AMS
Combinatorial analysis
Combinatorics
Combinatòria
Engineering Design
Equivalence
Geometry
Good drawings Empty triangles Erdos-Szekeres type problems
Graph theory
Graphs
Joining
Matemàtica discreta
Matemàtiques i estadística
Mathematics
Mathematics and Statistics
Original Paper
Texts
Triangles
Àrees temàtiques de la UPC
Erdős–Szekeres type problems
Good drawings
Empty triangles
ISSN:0911-0119, 1435-5914
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:A good drawing of a simple graph is a drawing on the sphere or, equivalently, in the plane in which vertices are drawn as distinct points, edges are drawn as Jordan arcs connecting their end vertices, and any pair of edges intersects at most once. In any good drawing, the edges of three pairwise connected vertices form a Jordan curve which we call a triangle. We say that a triangle is empty if one of the two connected components it induces does not contain any of the remaining vertices of the drawing of the graph. We show that the number of empty triangles in any good drawing of the complete graph K n with n vertices is at least n .
Bibliography:SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 14
ObjectType-Article-1
ObjectType-Feature-2
content type line 23
ISSN:0911-0119
1435-5914
DOI:10.1007/s00373-015-1550-5