1,133 research outputs found

    Compatible 4-Holes in Point Sets

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    Counting interior-disjoint empty convex polygons in a point set is a typical Erd\H{o}s-Szekeres-type problem. We study this problem for 4-gons. Let PP be a set of nn points in the plane and in general position. A subset QQ of PP, with four points, is called a 44-hole in PP if QQ is in convex position and its convex hull does not contain any point of PP in its interior. Two 4-holes in PP are compatible if their interiors are disjoint. We show that PP contains at least ⌊5n/11βŒ‹βˆ’1\lfloor 5n/11\rfloor {-} 1 pairwise compatible 4-holes. This improves the lower bound of 2⌊(nβˆ’2)/5βŒ‹2\lfloor(n-2)/5\rfloor which is implied by a result of Sakai and Urrutia (2007).Comment: 17 page

    Fullerenes with the maximum Clar number

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    The Clar number of a fullerene is the maximum number of independent resonant hexagons in the fullerene. It is known that the Clar number of a fullerene with n vertices is bounded above by [n/6]-2. We find that there are no fullerenes whose order n is congruent to 2 modulo 6 attaining this bound. In other words, the Clar number for a fullerene whose order n is congruent to 2 modulo 6 is bounded above by [n/6]-3. Moreover, we show that two experimentally produced fullerenes C80:1 (D5d) and C80:2 (D2) attain this bound. Finally, we present a graph-theoretical characterization for fullerenes, whose order n is congruent to 2 (respectively, 4) modulo 6, achieving the maximum Clar number [n/6]-3 (respectively, [n/6]-2)

    Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, II: The mixed Dirichlet-Neumann Problem

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    In this paper we continue the study started in part I (posted). We consider a planar, bounded, mm-connected region Ξ©\Omega, and let \bord\Omega be its boundary. Let T\mathcal{T} be a cellular decomposition of \Omega\cup\bord\Omega, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair (S,f)(S,f) where SS is a special type of a (possibly immersed) genus (mβˆ’1)(m-1) singular flat surface, tiled by rectangles and ff is an energy preserving mapping from T(1){\mathcal T}^{(1)} onto SS. In part I the solution of a Dirichlet problem defined on T(0){\mathcal T}^{(0)} was utilized, in this paper we employ the solution of a mixed Dirichlet-Neumann problem.Comment: 26 pages, 16 figures (color

    Cycle and Circle Tests of Balance in Gain Graphs: Forbidden Minors and Their Groups

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    We examine two criteria for balance of a gain graph, one based on binary cycles and one on circles. The graphs for which each criterion is valid depend on the set of allowed gain groups. The binary cycle test is invalid, except for forests, if any possible gain group has an element of odd order. Assuming all groups are allowed, or all abelian groups, or merely the cyclic group of order 3, we characterize, both constructively and by forbidden minors, the graphs for which the circle test is valid. It turns out that these three classes of groups have the same set of forbidden minors. The exact reason for the importance of the ternary cyclic group is not clear.Comment: 19 pages, 3 figures. Format: Latex2e. Changes: minor. To appear in Journal of Graph Theor

    Happy endings for flip graphs

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    We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include convex subsets of lattices, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.Comment: 26 pages, 15 figures. Revised and expanded for journal publicatio

    Bounds for the genus of a normal surface

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    This paper gives sharp linear bounds on the genus of a normal surface in a triangulated compact, orientable 3--manifold in terms of the quadrilaterals in its cell decomposition---different bounds arise from varying hypotheses on the surface or triangulation. Two applications of these bounds are given. First, the minimal triangulations of the product of a closed surface and the closed interval are determined. Second, an alternative approach to the realisation problem using normal surface theory is shown to be less powerful than its dual method using subcomplexes of polytopes.Comment: 38 pages, 25 figure
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