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  • Corners and simpliciality in oriented matroids and partial cubes
    Knauer, Kolja ; Marc, Tilen
    Building on a recent characterization of tope graphs of complexes of oriented matroids (COMs), we tackle and generalize several classical problems in oriented matroids (OMs), lopsided sets (aka ample ... set systems), and partial cubes via metric graph theory. Our first main result is that every element of an OM from a class introduced by Mandel is incident to a simplicial tope, i.e, such OMs contain no mutation-free elements. This allows us to refute a conjecture of Mandel from 1983, that would have implied the famous Las Vergnas’ simplex conjecture. Further, we show that the mutation graph of uniform OMs of order at most 9 are connected, thus confirming a stronger conjecture of Cordovil-Las Vergnas in this setting. The second main contribution is the introduction of corners of COMs as a natural generalization of corners in lopsided sets. Generalizing results of Bandelt and Chepoi, Tracy Hall, and Chalopin et al. we prove that realizable COMs, rank 2 COMs, as well as hypercellular graphs admit corner peelings. Using this, we confirm Las Vergnas’ conjecture for antipodal partial cubes (a class much larger than OMs) of small rank or isometric dimension.
    Vir: European journal of combinatorics = Journal européen de combinatoire = Europäische Zeitschrift für Kombinatorik. - ISSN 0195-6698 (Vol. 112, Aug. 2023, art. 103714 (27 str.))
    Vrsta gradiva - članek, sestavni del ; neleposlovje za odrasle
    Leto - 2023
    Jezik - angleški
    COBISS.SI-ID - 149966851

    Povezava(-e):

    https://www.sciencedirect.com/science/article/pii/S0195669823000318
    DOI

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