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...

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Veröffentlicht in:	Graphs and combinatorics Jg. 31; H. 2; S. 335 - 345
Hauptverfasser:	Aichholzer, Oswin, Hackl, Thomas, Pilz, Alexander, Ramos, Pedro, Sacristán, Vera, Vogtenhuber, Birgit
Format:	Journal Article Verlag
Sprache:	Englisch
Veröffentlicht:	Tokyo Springer Japan 01.03.2015 Springer Nature B.V
Schlagworte:	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-Zugang:	Volltext
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Zusammenfassung:	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 .
Bibliographie:	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