73 research outputs found

    Jungerman ladders and index 2 constructions for genus embeddings of dense regular graphs

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    We construct several families of genus embeddings of near-complete graphs using index 2 current graphs. In particular, we give the first known minimum genus embeddings of certain families of octahedral graphs, solving a longstanding conjecture of Jungerman and Ringel, and Hamiltonian cycle complements, making partial progress on a problem of White. Index 2 current graphs are also applied to various cases of the Map Color Theorem, in some cases yielding significantly simpler solutions, e.g., the nonorientable genus of K12s+8−K2K_{12s+8}-K_2. We give a complete description of the method, originally due to Jungerman, from which all these results were obtained.Comment: 23 pages, 21 figures; fixed 2 figures from previous versio

    Random Embeddings of Graphs: The Expected Number of Faces in Most Graphs is Logarithmic

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    A random 2-cell embedding of a connected graph GG in some orientable surface is obtained by choosing a random local rotation around each vertex. Under this setup, the number of faces or the genus of the corresponding 2-cell embedding becomes a random variable. Random embeddings of two particular graph classes -- those of a bouquet of nn loops and those of nn parallel edges connecting two vertices -- have been extensively studied and are well-understood. However, little is known about more general graphs despite their important connections with central problems in mainstream mathematics and in theoretical physics (see [Lando & Zvonkin, Springer 2004]). There are also tight connections with problems in computing (random generation, approximation algorithms). The results of this paper, in particular, explain why Monte Carlo methods (see, e.g., [Gross & Tucker, Ann. NY Acad. Sci 1979] and [Gross & Rieper, JGT 1991]) cannot work for approximating the minimum genus of graphs. In his breakthrough work ([Stahl, JCTB 1991] and a series of other papers), Stahl developed the foundation of "random topological graph theory". Most of his results have been unsurpassed until today. In our work, we analyze the expected number of faces of random embeddings (equivalently, the average genus) of a graph GG. It was very recently shown [Campion Loth & Mohar, arXiv 2022] that for any graph GG, the expected number of faces is at most linear. We show that the actual expected number of faces is usually much smaller. In particular, we prove the following results: 1) 12ln⁥n−2<E[F(Kn)]≀3.65ln⁥n\frac{1}{2}\ln n - 2 < \mathbb{E}[F(K_n)] \le 3.65\ln n, for nn sufficiently large. This greatly improves Stahl's n+ln⁥nn+\ln n upper bound for this case. 2) For random models B(n,Δ)B(n,\Delta) containing only graphs, whose maximum degree is at most Δ\Delta, we show that the expected number of faces is Θ(ln⁥n)\Theta(\ln n).Comment: 44 pages, 6 figure

    Master index of volumes 161–170

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    Genus Distributions of Cubic Outerplanar Graphs

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    Schematics of Graphs and Hypergraphs

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    Graphenzeichnen als ein Teilgebiet der Informatik befasst sich mit dem Ziel Graphen oder deren Verallgemeinerung Hypergraphen geometrisch zu realisieren. BeschrĂ€nkt man sich dabei auf visuelles Hervorheben von wesentlichen Informationen in Zeichenmodellen, spricht man von Schemata. Hauptinstrumente sind Konstruktionsalgorithmen und Charakterisierungen von Graphenklassen, die fĂŒr die Konstruktion geeignet sind. In dieser Arbeit werden Schemata fĂŒr Graphen und Hypergraphen formalisiert und mit den genannten Instrumenten untersucht. In der Dissertation wird zunĂ€chst das „partial edge drawing“ (kurz: PED) Modell fĂŒr Graphen (bezĂŒglich gradliniger Zeichnung) untersucht. Dabei wird um Kreuzungen im Zentrum der Kante visuell zu eliminieren jede Kante durch ein kreuzungsfreies TeilstĂŒck (= Stummel) am Start- und am Zielknoten ersetzt. Als Standard hat sich eine PED-Variante etabliert, in der das LĂ€ngenverhĂ€ltnis zwischen Stummel und Kante genau 1⁄4 ist (kurz: 1⁄4-SHPED). FĂŒr 1⁄4-SHPEDs werden Konstruktionsalgorithmen, Klassifizierung, Implementierung und Evaluation prĂ€sentiert. Außerdem werden PED-Varianten mit festen Knotenpositionen und auf Basis orthogonaler Zeichnungen erforscht. Danach wird das BUS Modell fĂŒr Hypergraphen untersucht, in welchem Hyperkanten durch fette horizontale oder vertikale – als BUS bezeichnete – Segmente reprĂ€sentiert werden. Dazu wird eine vollstĂ€ndige Charakterisierung von planaren Inzidenzgraphen von Hypergraphen angegeben, die eine planare Zeichnung im BUS Modell besitzen, und diverse planare BUS-Varianten mit festen Knotenpositionen werden diskutiert. Zum Schluss wird erstmals eine Punktmenge von subquadratischer GrĂ¶ĂŸe angegeben, die eine planare Einbettung (Knoten werden auf Punkte abgebildet) von 2-außenplanaren Graphen ermöglicht

    A new algorithm for recognizing the unknot

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    The topological underpinnings are presented for a new algorithm which answers the question: `Is a given knot the unknot?' The algorithm uses the braid foliation technology of Bennequin and of Birman and Menasco. The approach is to consider the knot as a closed braid, and to use the fact that a knot is unknotted if and only if it is the boundary of a disc with a combinatorial foliation. The main problems which are solved in this paper are: how to systematically enumerate combinatorial braid foliations of a disc; how to verify whether a combinatorial foliation can be realized by an embedded disc; how to find a word in the the braid group whose conjugacy class represents the boundary of the embedded disc; how to check whether the given knot is isotopic to one of the enumerated examples; and finally, how to know when we can stop checking and be sure that our example is not the unknot.Comment: 46 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol2/paper9.abs.htm
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