21 research outputs found

    The crossing number of locally twisted cubes

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    The {\it crossing number} of a graph GG is the minimum number of pairwise intersections of edges in a drawing of GG. Motivated by the recent work [Faria, L., Figueiredo, C.M.H. de, Sykora, O., Vrt'o, I.: An improved upper bound on the crossing number of the hypercube. J. Graph Theory {\bf 59}, 145--161 (2008)] which solves the upper bound conjecture on the crossing number of nn-dimensional hypercube proposed by Erd\H{o}s and Guy, we give upper and lower bounds of the crossing number of locally twisted cube, which is one of variants of hypercube.Comment: 17 pages, 12 figure

    Toward the Rectilinear Crossing Number of KnK_n: New Drawings, Upper Bounds, and Asymptotics

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    Scheinerman and Wilf (1994) assert that `an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph K_n.' A rectilinear drawing of K_n is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear, and that no three edges intersect in a point unless that point is an endpoint of all three. The rectilinear crossing number of K_n is the fewest number of edge crossings attainable over all rectilinear drawings of K_n. For each n we construct a rectilinear drawing of K_n that has the fewest number of edge crossings and the best asymptotics known to date. Moreover, we give some alternative infinite families of drawings of K_n with good asymptotics. Finally, we mention some old and new open problems.Comment: 13 Page

    The number of crossings in multigraphs with no empty lens

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    Let GG be a multigraph with nn vertices and e>4ne>4n edges, drawn in the plane such that any two parallel edges form a simple closed curve with at least one vertex in its interior and at least one vertex in its exterior. Pach and T\'oth (2018) extended the Crossing Lemma of Ajtai et al. (1982) and Leighton (1983) by showing that if no two adjacent edges cross and every pair of nonadjacent edges cross at most once, then the number of edge crossings in GG is at least αe3/n2\alpha e^3/n^2, for a suitable constant α>0\alpha>0. The situation turns out to be quite different if nonparallel edges are allowed to cross any number of times. It is proved that in this case the number of crossings in GG is at least αe2.5/n1.5\alpha e^{2.5}/n^{1.5}. The order of magnitude of this bound cannot be improved.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

    Degenerate Crossing Numbers

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    Let G be a graph with n vertices and e≥4n edges, drawn in the plane in such a way that if two or more edges (arcs) share an interior point p, then they properly cross one another at p. It is shown that the number of crossing points, counted without multiplicity, is at least constant timese and that the order of magnitude of this bound cannot be improved. If, in addition, two edges are allowed to cross only at most once, then the number of crossing points must exceed constant times(e/n)

    A successful concept for measuring non-planarity of graphs: the crossing number

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    AbstractThis paper surveys how the concept of crossing number, which used to be familiar only to a limited group of specialists, emerges as a significant graph parameter. This paper has dual purposes: first, it reviews foundational, historical, and philosophical issues of crossing numbers, second, it shows a new lower bound for crossing numbers. This new lower bound may be helpful in estimating crossing numbers
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