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Journal ArticleDOI

Polytopes, graphs, and complexes

01 Nov 1970-Bulletin of the American Mathematical Society (American Mathematical Society)-Vol. 76, Iss: 6, pp 1131-1201
About: This article is published in Bulletin of the American Mathematical Society.The article was published on 1970-11-01 and is currently open access. It has received 168 citations till now. The article focuses on the topics: Polytope.

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Citations
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17 Dec 1994
TL;DR: In this article, the Conjectures of Hadwiger and Hajos are used to define graph types, such as planar graph, graph on higher surfaces, and critical graph.
Abstract: Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.

1,380 citations

Journal ArticleDOI
TL;DR: For an arbitrary triangulated (d-1)-manifold without boundary, this paper showed that it is isomorphic to the boundary complex of a stacked polytope.
Abstract: For an arbitrary triangulated (d-1)-manifold without boundaryC withf 0 vertices andf 1 edges, define $$\gamma (C) = f_1 - df_0 + \left( {\begin{array}{*{20}c} {d + 1} \\ 2 \\ \end{array} } \right)$$ . Barnette proved that γ(C)≧0. We use the rigidity theory of frameworks and, in particular, results related to Cauchy's rigidity theorem for polytopes, to give another proof for this result. We prove that ford≧4, if γ(C)=0 thenC is a triangulated sphere and is isomorphic to the boundary complex of a stacked polytope. Other results: (a) We prove a lower bound, conjectured by Bjorner, for the number ofk-faces of a triangulated (d-1)-manifold with specified numbers of interior vertices and boundary vertices. (b) IfC is a simply connected triangulatedd-manifold,d≧4, and γ(lk(v, C))=0 for every vertexv ofC, then γ(C)=0. (lk(v,C) is the link ofv inC.) (c) LetC be a triangulatedd-manifold,d≧3. Then Ske11(Δ d+2) can be embedded in skel1 (C) iff γ(C)>0. (Δ d is thed-dimensional simplex.) (d) IfP is a 2-simpliciald-polytope then $$f_1 (P) \geqq df_0 (P) - \left( {\begin{array}{*{20}c} {d + 1} \\ 2 \\ \end{array} } \right)$$ . Related problems concerning pseudomanifolds, manifolds with boundary and polyhedral manifolds are discussed.

231 citations

Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of determining the possible f-vectors of simplicial polytopes and proved a conjecture about the form of the sclution to this problem.
Abstract: In this paper is considered the problem of determining the possiblef-vectors of simplicial polytopes. A conjecture is made about the form of the sclution to this problem; it is proved in the case ofd-polytopes with at mostd+3 vertices.

222 citations

Journal ArticleDOI
TL;DR: A theorem on paths with prescribed ends in a planar graph is proved which extends Tutte's theorem on cycles in planar graphs and implies the conjecture of Plummer asserting that every 4-connected planargraph is Hamiltonian-connected.
Abstract: We prove a theorem on paths with prescribed ends in a planar graph which extends Tutte's theorem on cycles in planar graphs [9] and implies the conjecture of Plummer [5] asserting that every 4-connected planar graph is Hamiltonian-connected.

159 citations

Journal ArticleDOI
TL;DR: The result stated in the title and a conjecture of Plummer that every graph which can be obtained from a 4-connected planar graph by deleting two vertices is Hamiltonian are proved.

146 citations

References
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Book
01 Jan 1960

2,165 citations

Journal ArticleDOI
TL;DR: In this paper, the authors define nodally 3-connected graphs as simple and non-separable graphs, and show how to obtain a convex representation of such graphs without Kuratowski subgraphs.
Abstract: W E use the definitions of (11). However, in deference to some recent attempts to unify the terminology of graph theory we replace the term 'circuit' by 'polygon', and 'degree' by 'valency'. A graph G is 3-connected (nodally 3-connected) if it is simple and non-separable and satisfies the following condition; if G is the union of two proper subgraphs H and K such that HnK consists solely of two vertices u and v, then one of H and K is a link-graph (arc-graph) with ends u and v. It should be noted that the union of two proper subgraphs H and K of G can be the whole of G only if each of H and K includes at least one edge or vertex not belonging to the other. In this paper we are concerned mainly with nodally 3-connected graphs, but a specialization to 3-connected graphs is made in § 12. In § 3 we discuss conditions for a nodally 3-connected graph to be planar, and in § 5 we discuss conditions for the existence of Kuratowski subgraphs of a given graph. In §§ 6-9 we show how to obtain a convex representation of a nodally 3-connected graph, without Kuratowski subgraphs, by solving a set of linear equations. Some extensions of these results to general graphs, with a proof of Kuratowski's theorem, are given in §§ 10-11. In § 12 we discuss the representation in the plane of a pair of dual graphs, and in § 13 we draw attention to some unsolved problems.

1,124 citations

BookDOI
01 Jan 1957
TL;DR: In this article, a vorliegende Buch vereinigt in seinen wesentlichen Teilen den Stoff verschiedener Vorlesungen liber Inhaltstheorie, isoperimetrische Probleme and liber konvexe Karper and allgemeine Integralgeometrie, which ich im Laufe der letten yearre an der Universitat Bern gehalten habe.
Abstract: Das vorliegende Buch vereinigt in seinen wesentlichen Teilen den Stoff verschiedener Vorlesungen liber Inhaltstheorie, isoperimetrische Probleme und liber konvexe Karper und allgemeine Integralgeometrie, die ich im Laufe der letzten Jahre an der Universitat Bern gehalten habe. Abgesehen von einzelnen kleinen Spezialvorlesungen entsprachen die Kurse dem Lehrprogramm fUr die allgemeine EinfUhrung in die hahere Mathematik und waren demnach fUr Harer der unteren und mittleren Semester bestimmt. Bei der buchmaBigen Zusammenfassung war ich bemliht, in einer sich auf Stoffauswahl und Behandlungsart auswirkenden Ausrichtung auf die Linie der elementaren direkten Mengengeometrie die gemeinsame Bindung zu finden, welche die verschiedenartigen Sachgebiete, die auch unabhangig durchfUhrbaren Vorlesungen entsprechen, zu einem einheit- lichen Ganzen zusammenfUgen solI. Mit der erwahnten Beschrankung wurde eine einfache, ohne hahere Spezialkenntnisse lesbare Darstellung der einschlagigen Themen erzielt. Erforderlich sind gute Kenntnisse der Grundtatsachen der Elementargeometrie, eine gewisse Vertrautheit mit den wichtigsten Begriffen der Punktmengenlehre und mit der mengentheoretischen SchluBweise, schlieBlich einige Dbung beim Umgang mit exakten, sich auf Raum und Zahl beziehende Begriffs- bildungen. Welches sind nun die Kriterien einer elementaren und direkt mengen- geometrischen Betrachtungsweise, wie sie dem vorliegenden Buche zugrunde liegen solI? Einige hierfUr charakteristische Merkmale seien nachfolgend aufgezahlt: 1. Alles spielt sich einheitlich im (k-dimensionalen) euklidischen Raum ab; so ist ein unveranderliches Arbeitsfeld gegeben, mit dem man von den Elementen her gut vertraut ist.

1,094 citations