论文标题
角落和简单性的矩形和部分立方体
Corners and simpliciality in oriented matroids and partial cubes
论文作者
论文摘要
我们以最新的表征对定向矩阵（COM）的toper图进行了表征，我们通过度量图理论解决并推广几个经典问题（OMS），偏斜的集合（又称足够的集合系统）和部分立方体。我们的第一个主要结果是，曼德尔引入的类别中的OM的每个元素都出现在简单的TOPE上，即，此类OMS不包含无突变元素。这使我们能够从1983年起反驳曼德尔的猜想，这意味着著名的Las Vergnas的单纯猜想。此外，我们表明，最多9个均匀OM的突变图是连接的，因此在这种情况下确认了Cordovil-Las Vergnas的更强有力的猜想。 第二个主要贡献是引入COM的角落，作为在偏斜的集合中的自然概括。 Bandelt和Chepoi，Tracy Hall和Chalopin等人的概括。我们证明，可实现的COM，排名2 coms以及高细胞图允许角脱皮。使用此情况，我们确认了对小等级或等距维度的抗虫部分立方体（比OMS大多数的类别）的LAS Vergnas的猜想。
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 lager than OMs) of small rank or isometric dimension.
