Randomized Construction of Complexes with Large Diameter

Francisco Criado, Andrew Newman

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.