Abstract
In their paper proving the Hirsch bound for flag normal simplicial complexes, Adiprasito and Benedetti (Math Oper Res 39(4):1340–1348, 2014) define the notion of combinatorial segment. The study of the maximal length of these objects provides the upper bound \(O(n2^d)\) for the diameter of any normal pure simplicial complex of dimension d with n vertices, and the Hirsch bound \(n-d+1\) if the complexes are moreover flag. In the present article, we propose a formulation of combinatorial segments which is equivalent but more local, by introducing the notions of monotonicity and conservativeness of dual paths in pure simplicial complexes. We use these definitions to investigate further properties of combinatorial segments. Besides recovering the two stated bounds, we show a refined bound for banner complexes, and study the behavior of the maximal length of combinatorial segments with respect to two usual operations, namely join and one-point suspension. Finally, we show the limitations of combinatorial segments by constructing pure normal simplicial complexes in which all combinatorial segments between two particular facets achieve the length \(\Omega (n2^{d})\). This includes vertex-decomposable—therefore Hirsch—polytopes.
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We thank the anonymous referees for their valuable comments and advice in improving the clarity and presentation of the article.
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Jean-Philippe Labbé: With the support of a FQRNT post-doctoral fellowship and a post-doctoral ISF Grant (805/11). Thibault Manneville: With the support of an École Polytechnique Gaspard Monge doctoral grant and with partial support of French ANR Grant EGOS (12 JS02 002 01). Francisco Santos: With the support of the Spanish Ministry of Science through Grants MTM2011-22792 and MTM2014-54207-P and of the Einstein Foundation Berlin.
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Labbé, JP., Manneville, T. & Santos, F. Hirsch polytopes with exponentially long combinatorial segments. Math. Program. 165, 663–688 (2017). https://doi.org/10.1007/s10107-016-1099-y
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DOI: https://doi.org/10.1007/s10107-016-1099-y