29 research outputs found

    Removing Even Crossings

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    An edge in a drawing of a graph is called even\textit{even} if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (not using Kuratowski's theorem), and the result that the odd crossing number of a graph equals the crossing number of the graph for values of at most 33. We begin with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte

    Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4

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    We find a graph of genus 55 and its drawing on the orientable surface of genus 44 with every pair of independent edges crossing an even number of times. This shows that the strong Hanani-Tutte theorem cannot be extended to the orientable surface of genus 44. As a base step in the construction we use a counterexample to an extension of the unified Hanani-Tutte theorem on the torus.Comment: 12 pages, 4 figures; minor revision, new section on open problem

    On the size of planarly connected crossing graphs

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    We prove that if an nn-vertex graph GG can be drawn in the plane such that each pair of crossing edges is independent and there is a crossing-free edge that connects their endpoints, then GG has O(n)O(n) edges. Graphs that admit such drawings are related to quasi-planar graphs and to maximal 11-planar and fan-planar graphs.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

    On the optimality of the Arf invariant formula for graph polynomials

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    We prove optimality of the Arf invariant formula for the generating function of even subgraphs, or, equivalently, the Ising partition function, of a graph.Comment: Advances in Mathematics, 201

    Strong Hanani-Tutte on the Projective Plane

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    If a graph can be drawn in the projective plane so that every two non-adjacent edges cross an even number of times, then the graph can be embedded in the projective plane

    Adjacent Crossings Do Matter

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    Strong Hanani-Tutte for the Torus

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    If a graph can be drawn on the torus so that every two independent edges cross an even number of times, then the graph can be embedded on the torus

    The Z2\mathbb{Z}_2-genus of Kuratowski minors

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    A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The Z2\mathbb{Z}_2-genus of a graph GG is the minimum gg such that GG has an independently even drawing on the orientable surface of genus gg. An unpublished result by Robertson and Seymour implies that for every tt, every graph of sufficiently large genus contains as a minor a projective t×tt\times t grid or one of the following so-called tt-Kuratowski graphs: K3,tK_{3,t}, or tt copies of K5K_5 or K3,3K_{3,3} sharing at most 22 common vertices. We show that the Z2\mathbb{Z}_2-genus of graphs in these families is unbounded in tt; in fact, equal to their genus. Together, this implies that the genus of a graph is bounded from above by a function of its Z2\mathbb{Z}_2-genus, solving a problem posed by Schaefer and \v{S}tefankovi\v{c}, and giving an approximate version of the Hanani-Tutte theorem on orientable surfaces. We also obtain an analogous result for Euler genus and Euler Z2\mathbb{Z}_2-genus of graphs.Comment: 23 pages, 7 figures; a few references added and correcte

    Hanani-Tutte for radial planarity

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    A drawing of a graph G is radial if the vertices of G are placed on concentric circles C 1 , . . . , C k with common center c , and edges are drawn radially : every edge intersects every circle centered at c at most once. G is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of G are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling. We show that a graph G is radial planar if G has a radial drawing in which every two edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the weak variant of the Hanani-Tutte theorem for radial planarity. This generalizes a result by Pach and Toth
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