44,424 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 TĂłth 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

    Removing Independently Even Crossings

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    Unexpected behaviour of crossing sequences

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    The n-th crossing number of a graph G, denoted cr_n(G), is the minimum number of crossings in a drawing of G on an orientable surface of genus n. We prove that for every a>b>0, there exists a graph G for which cr_0(G) = a, cr_1(G) = b, and cr_2(G) = 0. This provides support for a conjecture of Archdeacon et al. and resolves a problem of Salazar.Comment: 21 page

    The forbidden number of a knot

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    Every classical or virtual knot is equivalent to the unknot via a sequence of extended Reidemeister moves and the so-called forbidden moves. The minimum number of forbidden moves necessary to unknot a given knot is an invariant we call the {\it forbidden number}. We relate the forbidden number to several known invariants, and calculate bounds for some classes of virtual knots.Comment: 14 pages, many figures; v2 improves the upper bounds from the crossing number, and adds more detail to the data presented in the conclusio

    Simple realizability of complete abstract topological graphs simplified

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    An abstract topological graph (briefly an AT-graph) is a pair A=(G,X)A=(G,\mathcal{X}) where G=(V,E)G=(V,E) is a graph and X⊆(E2)\mathcal{X}\subseteq {E \choose 2} is a set of pairs of its edges. The AT-graph AA is simply realizable if GG can be drawn in the plane so that each pair of edges from X\mathcal{X} crosses exactly once and no other pair crosses. We show that simply realizable complete AT-graphs are characterized by a finite set of forbidden AT-subgraphs, each with at most six vertices. This implies a straightforward polynomial algorithm for testing simple realizability of complete AT-graphs, which simplifies a previous algorithm by the author. We also show an analogous result for independent Z2\mathbb{Z}_2-realizability, where only the parity of the number of crossings for each pair of independent edges is specified.Comment: 26 pages, 17 figures; major revision; original Section 5 removed and will be included in another pape
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