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
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Abstract	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 .
AbstractList	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 . (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) Issue Title: Special Issue: Combinatorics and Geometry 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 ... with ... vertices is at least ... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).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 ... with ... vertices is at least ... 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 Kn with n vertices is at least n. Peer Reviewed
Author	Vogtenhuber, Birgit Hackl, Thomas Pilz, Alexander Sacristán, Vera Aichholzer, Oswin Ramos, Pedro
Author_xml	– sequence: 1 givenname: Oswin surname: Aichholzer fullname: Aichholzer, Oswin organization: Institute for Software Technology, Graz University of Technology – sequence: 2 givenname: Thomas surname: Hackl fullname: Hackl, Thomas organization: Institute for Software Technology, Graz University of Technology – sequence: 3 givenname: Alexander surname: Pilz fullname: Pilz, Alexander organization: Institute for Software Technology, Graz University of Technology – sequence: 4 givenname: Pedro surname: Ramos fullname: Ramos, Pedro organization: Departamento de Matemáticas, Universidad de Alcalá – sequence: 5 givenname: Vera surname: Sacristán fullname: Sacristán, Vera organization: Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya – sequence: 6 givenname: Birgit surname: Vogtenhuber fullname: Vogtenhuber, Birgit email: bvogt@ist.tugraz.at organization: Institute for Software Technology, Graz University of Technology
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Cites_doi	10.1002/jgt.20249 10.1016/S0012-365X(98)00098-3 10.1007/s00454-007-1343-6 10.1007/s00454-010-9320-x 10.1109/SFCS.1977.21 10.1007/BF01903339 10.1145/2462356.2462394 10.1007/s00454-007-9018-x 10.2307/2974980 10.4153/CMB-1983-077-8
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Contributor	Universitat Politècnica de Catalunya. DCCG - Grup de recerca en geometria computacional, combinatoria i discreta Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II
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References	Fulek, R., Ruiz-Vargas, A.J.: Topological graphs: empty triangles and disjoint matchings. In: Proceedings of 29th ACM Symposium on Computational Geometry (SoCG’13), Rio de Janeiro, Brazil, pp. 259–265 (2013) Lloyd, E.L.: On triangulations of a set of points in the plane. In: 18th Annual Symposium on Foundations of Computer Science, pp. 228–240 (1977) KatchalskiMMeirAOn empty triangles determined by points in the planeActa Math. Hung.1988513–432332810.1007/BF019033390655.52007956984 RichterRBThomassenCRelations between crossing numbers of complete and complete bipartite graphsAm. Math. Mon.1997104213113710.2307/29749800872.050101437414 HortonJSets with no empty convex 7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$7$$\end{document}-gonsCan. Math. Bull.198326448248410.4153/CMB-1983-077-80521.52010716589 HarborthHEmpty triangles in drawings of the complete graphDiscrete Math.199819110911110.1016/S0012-365X(98)00098-30956.050341644877 PanSRichterRBThe crossing number of K11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{11}$$\end{document} is 100J. Graph Theory200756212813410.1002/jgt.202491128.050182350621 BárányIValtrPPlanar point sets with a small number of empty convex polygonsStud. Sci. Math. Hung.20044122432661075.52008 ErdősPSome more problems on elementary geometryAust. Math. Soc. Gaz.197855254 Aichholzer, O., Fabila-Monroy, R., Hackl, T., Huemer, C., Pilz, A., Vogtenhuber, B.: Lower bounds for the number of small convex k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-holes. In: Proceedings of 24th Canadian Conference on Computational Geometry CCCG’12, Charlottetown, Canada, pp. 247–252. (2012) GerkenTEmpty convex hexagons in planar point setsDiscrete Comput. Geom.2008391–323927210.1007/s00454-007-9018-x1184.520162383761 HarborthHKonvexe Fünfecke in ebenen PunktmengenElemente der Mathematik1978331161180397.52005510092In German KynčlJSimple realizability of complete abstract topological graphs in PDiscrete Comput. Geom.201145338339910.1007/s00454-010-9320-x1214.050132770542 NicolásCThe empty hexagon theoremDiscrete Comput. Geom.200738238939710.1007/s00454-007-1343-61146.520102343313 1550_CR4 RB Richter (1550_CR14) 1997; 104 T Gerken (1550_CR5) 2008; 39 S Pan (1550_CR13) 2007; 56 1550_CR1 H Harborth (1550_CR7) 1998; 191 J Horton (1550_CR8) 1983; 26 J Kynčl (1550_CR10) 2011; 45 1550_CR11 C Nicolás (1550_CR12) 2007; 38 M Katchalski (1550_CR9) 1988; 51 I Bárány (1550_CR2) 2004; 41 H Harborth (1550_CR6) 1978; 33 P Erdős (1550_CR3) 1978; 5
References_xml	– reference: Fulek, R., Ruiz-Vargas, A.J.: Topological graphs: empty triangles and disjoint matchings. In: Proceedings of 29th ACM Symposium on Computational Geometry (SoCG’13), Rio de Janeiro, Brazil, pp. 259–265 (2013) – reference: HortonJSets with no empty convex 7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$7$$\end{document}-gonsCan. Math. Bull.198326448248410.4153/CMB-1983-077-80521.52010716589 – reference: KatchalskiMMeirAOn empty triangles determined by points in the planeActa Math. Hung.1988513–432332810.1007/BF019033390655.52007956984 – reference: Lloyd, E.L.: On triangulations of a set of points in the plane. In: 18th Annual Symposium on Foundations of Computer Science, pp. 228–240 (1977) – reference: BárányIValtrPPlanar point sets with a small number of empty convex polygonsStud. Sci. Math. Hung.20044122432661075.52008 – reference: NicolásCThe empty hexagon theoremDiscrete Comput. Geom.200738238939710.1007/s00454-007-1343-61146.520102343313 – reference: GerkenTEmpty convex hexagons in planar point setsDiscrete Comput. Geom.2008391–323927210.1007/s00454-007-9018-x1184.520162383761 – reference: HarborthHKonvexe Fünfecke in ebenen PunktmengenElemente der Mathematik1978331161180397.52005510092In German – reference: Aichholzer, O., Fabila-Monroy, R., Hackl, T., Huemer, C., Pilz, A., Vogtenhuber, B.: Lower bounds for the number of small convex k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-holes. In: Proceedings of 24th Canadian Conference on Computational Geometry CCCG’12, Charlottetown, Canada, pp. 247–252. (2012) – reference: ErdősPSome more problems on elementary geometryAust. Math. Soc. Gaz.197855254 – reference: KynčlJSimple realizability of complete abstract topological graphs in PDiscrete Comput. Geom.201145338339910.1007/s00454-010-9320-x1214.050132770542 – reference: HarborthHEmpty triangles in drawings of the complete graphDiscrete Math.199819110911110.1016/S0012-365X(98)00098-30956.050341644877 – reference: RichterRBThomassenCRelations between crossing numbers of complete and complete bipartite graphsAm. Math. Mon.1997104213113710.2307/29749800872.050101437414 – reference: PanSRichterRBThe crossing number of K11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{11}$$\end{document} is 100J. Graph Theory200756212813410.1002/jgt.202491128.050182350621 – volume: 56 start-page: 128 issue: 2 year: 2007 ident: 1550_CR13 publication-title: J. Graph Theory doi: 10.1002/jgt.20249 – volume: 191 start-page: 109 year: 1998 ident: 1550_CR7 publication-title: Discrete Math. doi: 10.1016/S0012-365X(98)00098-3 – volume: 38 start-page: 389 issue: 2 year: 2007 ident: 1550_CR12 publication-title: Discrete Comput. Geom. doi: 10.1007/s00454-007-1343-6 – volume: 45 start-page: 383 issue: 3 year: 2011 ident: 1550_CR10 publication-title: Discrete Comput. Geom. doi: 10.1007/s00454-010-9320-x – ident: 1550_CR1 – volume: 33 start-page: 116 year: 1978 ident: 1550_CR6 publication-title: Elemente der Mathematik – volume: 5 start-page: 52 year: 1978 ident: 1550_CR3 publication-title: Aust. Math. Soc. Gaz. – ident: 1550_CR11 doi: 10.1109/SFCS.1977.21 – volume: 41 start-page: 243 issue: 2 year: 2004 ident: 1550_CR2 publication-title: Stud. Sci. Math. Hung. – volume: 51 start-page: 323 issue: 3–4 year: 1988 ident: 1550_CR9 publication-title: Acta Math. Hung. doi: 10.1007/BF01903339 – ident: 1550_CR4 doi: 10.1145/2462356.2462394 – volume: 39 start-page: 239 issue: 1–3 year: 2008 ident: 1550_CR5 publication-title: Discrete Comput. Geom. doi: 10.1007/s00454-007-9018-x – volume: 104 start-page: 131 issue: 2 year: 1997 ident: 1550_CR14 publication-title: Am. Math. Mon. doi: 10.2307/2974980 – volume: 26 start-page: 482 issue: 4 year: 1983 ident: 1550_CR8 publication-title: Can. Math. Bull. doi: 10.4153/CMB-1983-077-8
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SubjectTerms	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
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Title	Empty Triangles in Good Drawings of the Complete Graph
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