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Connected Hyperplanes in Binary Matroids

✍ Scribed by Jennifer McNulty; Haidong Wu

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 125 KB
Volume: 79
Category: Article
ISSN: 0095-8956
DOI: 10.1006/jctb.1999.1949

No coin nor oath required. For personal study only.

✦ Synopsis

In this paper, we prove that any simple and cosimple connected binary matroid has at least four connected hyperplanes. We further prove that each element in such a matroid is contained in at least two connected hyperplanes. Our main result generalizes a matroid result of Kelmans, and independently, of Seymour. The following consequence of the main result generalizes a graph result of Thomassen and Toft on induced non-separating cycles and another graph result of Kaugars on deletable vertices. If G is a simple 2-connected graph with minimum degree at least 3, then, for every edge e, there are at least two induced non-separating cycles avoiding e and two deletable vertices non-incident to e. Moreover, G has at least four induced non-separating cycles.

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