Abstract
We construct d-dimensional pure simplicial complexes and pseudo-manifolds (without boundary) with n vertices whose combinatorial diameter grows as \(c_d n^{d-1}\) for a constant \(c_d\) depending only on d, which is the maximum possible growth. Moreover, the constant \(c_d\) is optimal modulo a singly exponential factor in d. The pure simplicial complexes improve on a construction of the second author that achieved \(c_d n^{2d/3}\). For pseudo-manifolds without boundary, as far as we know, no construction with diameter greater than \(n^2\) was previously known.
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Notes
The standard usage is to consider all faces, not only facets, as elements of C, and then call facets the maximal ones. Our approach is equivalent and, for our purposes, simpler.
C is a pseudo-manifold with boundary if ridges are contained in at most two facets, the boundary of C consisting of the ridges lying in only one facet. Standard usage is to say “pseudo-manifold” alone meaning “without boundary” and “pseudo-manifold with boundary” when boundary is allowed. But to avoid confusion we here insist in saying “without boundary” when boundary is forbidden.
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Acknowledgements
Work of Francisco Santos is supported in part by the Spanish Ministry of Science (MICINN) through Grant MTM2014-54207P.
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Editor in Charge: Günter M. Ziegler
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Criado, F., Santos, F. The Maximum Diameter of Pure Simplicial Complexes and Pseudo-manifolds. Discrete Comput Geom 58, 643–649 (2017). https://doi.org/10.1007/s00454-017-9888-5
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DOI: https://doi.org/10.1007/s00454-017-9888-5