The Holt-Klee condition for oriented matroids

Komei Fukuda, Sonoko Moriyama, Yoshio Okamoto

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)

Abstract

Holt and Klee have recently shown that every (generic) LP orientation of the graph of a d-polytope satisfies a directed version of the d-connectivity property, i.e. there are d internally disjoint directed paths from a unique source to a unique sink. We introduce two new classes HK and HK* of oriented matroids (OMs) by enforcing this property and its dual interpretation in terms of line shellings, respectively. Both classes contain all representable OMs by the Holt-Klee theorem. While we give a construction of an infinite family of non-HK* OMs, it is not clear whether there exists any non-HK OM. This leads to a fundamental question as to whether the Holt-Klee theorem can be proven combinatorially by using the OM axioms only. Finally, we give the complete classification of OM(4, 8), the OMs of rank 4 on 8-element ground set with respect to the HK, HK*, Euclidean and Shannon properties. Our classification shows that there exists no non-HK OM in this class.

Original languageEnglish
Pages (from-to)1854-1867
Number of pages14
JournalEuropean Journal of Combinatorics
Volume30
Issue number8
DOIs
Publication statusPublished - 2009 Nov

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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