Convexity and Discrete Geometry Including Graph Theory: Mulhouse, France, September 2014 PDF
Preview Convexity and Discrete Geometry Including Graph Theory: Mulhouse, France, September 2014
Springer Proceedings in Mathematics & Statistics Karim Adiprasito Imre Bárány Costin Vîlcu E ditors Convexity and Discrete Geometry Including Graph Theory Mulhouse, France, September 2014 Springer Proceedings in Mathematics & Statistics Volume 148 Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. More information about this series at http://www.springer.com/series/10533 á á Karim Adiprasito Imre B r ny (cid:129) î Costin V lcu Editors Convexity and Discrete Geometry Including Graph Theory Mulhouse, France, September 2014 123 Editors Karim Adiprasito Costin Vîlcu Einstein Institute for Mathematics “Simion Stoilow”Institute ofMathematics HebrewUniversity of Jerusalem ofthe Roumanian Academy Jerusalem Bucharest Israel Roumania Imre Bárány RényiInstitute ofMathematics Hungarian Academy of Sciences Budapest Hungary and Department of Mathematics University CollegeLondon London UK ISSN 2194-1009 ISSN 2194-1017 (electronic) SpringerProceedings in Mathematics& Statistics ISBN978-3-319-28184-1 ISBN978-3-319-28186-5 (eBook) DOI 10.1007/978-3-319-28186-5 LibraryofCongressControlNumber:2015959922 MathematicsSubjectClassification: 52-XX,51-XX,05-XX,68R10 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Contents Part I Research Articles Tudor Zamfirescu: From Convex to Magic . . . . . . . . . . . . . . . . . . . . . 3 Solomon Marcus Transformations of Digraphs Viewed as Intersection Digraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Christina M.D. Zamfirescu Acute Triangulations of Rectangles, with Angles Bounded Below. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Liping Yuan Multi-compositions in Exponential Counting of Hypohamiltonian Snarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Zdzisław Skupień Hamiltonicity in k-tree-Halin Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . 59 Ayesha Shabbir and Tudor Zamfirescu Reflections of Planar Convex Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Rolf Schneider Steinhaus Conditions for Convex Polyhedra. . . . . . . . . . . . . . . . . . . . . 77 Joël Rouyer About the Hausdorff Dimension of the Set of Endpoints of Convex Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Alain Rivière About a Surprising Computer Program of Matthias Müller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Mihai Prunescu On the Connected Spanning Cubic Subgraph Problem. . . . . . . . . . . . . 109 Damien Massé, Reinhardt Euler and Laurent Lemarchand v vi Contents Extremal Results on Intersection Graphs of Boxes in Rd . . . . . . . . . . . 137 Alvaro Martínez-Pérez, Luis Montejano and Deborah Oliveros On the Helly Dimension of Hanner Polytopes. . . . . . . . . . . . . . . . . . . . 145 János Kincses T(4) Families of ϕ-Disjoint Ovals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Aladár Heppes and Jesús Jerónimo-Castro Fair Partitioning by Straight Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Augustin Fruchard and Alexander Magazinov Fixed Point Theorems for Multivalued Zamfirescu Operators in Convex Kasahara Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Alexandru-Darius Filip and Adrian Petruşel Complex Conference Matrices, Complex Hadamard Matrices and Complex Equiangular Tight Frames . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Boumediene Et-Taoui Envelopes of α-Sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Nicolas Chevallier, Augustin Fruchard and Costin Vîlcu Selected Open and Solved Problems in Computational Synthetic Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Jürgen Bokowski, Jurij Kovič, Tomaž Pisanski and Arjana Žitnik Reductions of 3-Connected Quadrangulations of the Sphere. . . . . . . . . 231 Sheng Bau Paths on the Sphere Without Small Angles . . . . . . . . . . . . . . . . . . . . . 239 Imre Bárány and Attila Pór Part II Open Problem Notes Seven Problems on Hypohamiltonian and Almost Hypohamiltonian Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Carol T. Zamfirescu Six Problems on the Length of the Cut Locus . . . . . . . . . . . . . . . . . . . 257 Costin Vîlcu and Tudor Zamfirescu An Existence Problem for Matroidal Families . . . . . . . . . . . . . . . . . . . 261 José Manuel dos Santos Simões-Pereira Two Problems on Cages for Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Luis Montejano and Tudor Zamfirescu Problem Session: Cubical Pachner Moves . . . . . . . . . . . . . . . . . . . . . . 265 Louis Funar Contents vii Problems in Discrete Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Jürgen Eckhoff What Is the Minimal Cardinal of a Family Which Shatters All d-Subsets of a Finite Set?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Nicolas Chevallier and Augustin Fruchard Some Open Problems of Ramsey Minimal Graphs. . . . . . . . . . . . . . . . 279 Edy Tri Baskoro Introduction This volume is dedicated to Tudor Zamfirescu, on the occasion of his 70th anniversary.TherewasaconferencecelebratingthesameanniversaryatMulhouse, France, during 7–11 September 2014, held with financial support from local communities and the Foundation Compositio Mathematica. The idea of creating this book emerged there and was met with enthusiastic support from the participants. Tudor Zamfirescu’s mathematics is rich and broad and unique. He is neither a fan of Bourbaki nor an inventor of some general theories. He is a problem solver and, equally and importantly, a problem poser. For him, according to his own words, “mathematics was and is a world of individual strange objects, waiting for our understanding, but proving to be, in most cases, in conflict with our intuition and expectations”. Tudor likes problems that are easy to understand but whose solution requires serious effort, especially when the solution turns out to be sur- prising, counterintuitive and aesthetic. Just to mention a few, the results in his papers “Most convex mirrors are magic” or “Every point is critical” or “Many endpoints and few interior points of geodesics” belong to this category. This volume wants to continue this tradition. It presents easily understandable butsurprisingproperties,obtainedusingtopological,geometricandgraphtheoretic tools in convexity (in geometry or analysis) and discrete geometry (including here graphtheory).Tudorhadmanycollaboratorsalonghisbroadandrichmathematical activity, and this volume has many contributors. Tudor likes to solve—and to propose—openproblems.Thefirstpartofthisvolumeconsistsofresearcharticles, while the second part offers an exciting bouquet of open problems. Written by top experts, the contributions underline the intimate connections between thevariousfields,theconnectionstootherdomainsofgeometryandtheir reciprocal influence. They propose thereader an overview on recent developments in geometry and at its border with discrete mathematics, and provide many open questions. The volume is intended for a large audience in mathematics, including researchers and graduate students interested in geometry and graph theory. Just for a change, the articles and open problems are listed in reverse alpha- betical order of the authors’ names. We make one exception to this rule: we start ix x Introduction withthepaperbySolomonMarcus,aformerProfessorofTudor.Hiswasoneofthe most appreciated talksat Mulhouse: it described TudorZamfirescu’s mathematical lifeandachievements,includingmanypersonalaspects,andalsoreflectedthrough the vast mathematical and philosophical culture of the author. Marcus’ paper is a written version of his talk at Mulhouse. In the following we briefly underline background connections between the papers in this volume. These rough explanations may, of course, be completed by the readers. Properties of complete metric spaces, convex surfaces and geodesics are treated in the articles by Alexandru-Darius Filip and Adrian Petruşel, and by AlainRivière,andbyJoëlRouyer.Therearestillexcitingopenquestionsinplanar convex geometry, and such problems are considered in the papers of Rolf Schneider,ofAladárHeppesandJesúsJerónimo-Castro,andofNicolasChevallier, Augustin Fruchard and Costin Vîlcu. Similar is the topic of the contribution by Augustin Fruchard and Alexander Magazinov. János Kincses improves the known bounds for the Helly dimension of the L -sum of centrally symmetric compact 1 convex bodies. Combinatorial properties of boxes are investigated by Alvaro Martnez-Pérez, Luis Montejano Peimbert and Deborah Oliveros Braniff. Jürgen Bokowski, Jurij Kovič, Tomaž Pisanski and Arjana Žitnik write about geometric realizations of combinatorial structures. Imre Bárány and Attila Pór solve a metric/combinatorial problem on the two-dimensional sphere. Triangulations and quadrangulations are treated in the articles of Liping Yuan and of Sheng Bau. Graphtheory,especiallyHamiltoncyclesandpathsingraphs,havealwaysbeenof definite interest for Tudor Zamfirescu. This is reflected in several contributions. TudorZamfirescutogetherwithAyeshaShabbirinvestigatestheHamiltonicityand traceability of k-tree-Halin graphs. Hypohamiltonian snarks are the topic of the paper by Zdzisław Skupień. The article by Christina Zamfirescu is about inter- sectiongraphsanddigraphs.Thespanningcubicsubgraphproblemanditsrelatives are the topic in the paper by Damien Massé, Reinhardt Euler and Laurent Lemarchand.GraphsareemployedbyMihaiPrunescuforthestudyofasurprising algorithm. Boumediene Et-Taoui constructs highly symmetric complex matrices. We wish Tudor many happy returns and many more beautiful theorems. Jerusalem Karim Adiprasito Budapest Imre Bárány Bucharest CostinVîlcu August 2015
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