Abstract
We consider the question of the largest possible combinatorial diameter among pure dimensional and strongly connected (d-1)-dimensional simplicial complexes on n vertices, denoted H_s(n, d). Using a probabilistic construction we give a new lower bound on H_s(n, d) that is within an O(d^2) factor of the upper bound. This improves on the previously best known lower bound which was within a factor of e^{varTheta (d)} of the upper bound. We also make a similar improvement in the case of pseudomanifolds.
Highlights
Given a pure simplicial complex C, one may define the dual graph G(C) as the graph whose vertices are the top-dimensional faces of C and whose edges are pairs of top-dimensional faces that intersect at a face of codimension 1
Using standard terminology we refer to top-dimensional faces of C as facets and codimension 1 faces as ridges
In the probabilistic step we take a quotient of the complex that preserves the diameter, drops the number of vertices to Θ(L1/(d−1)), and remains a simplicial complex
Summary
Given a pure simplicial complex C, one may define the dual graph G(C) as the graph whose vertices are the top-dimensional faces of C and whose edges are pairs of top-dimensional faces that intersect at a face of codimension 1. Following the notation of [3], we define Hs(n, d) to be the largest combinatorial diameter of any strongly connected, pure (d − 1)-dimensional simplicial complex on n vertices. In the probabilistic step we take a quotient of the complex that preserves the diameter, drops the number of vertices to Θ(L1/(d−1)), and remains a simplicial complex This approach works for the class of pseudomanifolds, which is considered in [3]. A pseudomanifold without boundary is a pure dimensional and strongly connected simplicial complex so that every ridge is contained in exactly two facets. A result of [3] shows that Hpm(n, d) = Θd (nd−1), but again the ratio between the upper bound and the lower bound is eΘ(d) We improve this to Θ(d3): Theorem 1.3 Fix d ≥ 3, Hpm(n, d) satisfies. In this case we slightly improve on the upper bound too by using the fact that G(C) is d-regular when C is a (d − 1)-pseudomanifold
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