The Maximum Diameter of 2-Dimensional Simplicial Complexes

Abstract We study a problem of Santos about the largest possible diameter of a d -dimensional (abstract) simplicial complex on n vertices. For dimension 2, we determine the exact value of the maximum for every n using an explicit construction. We also come across a tantalizing open problem about the packing of squares of Hamilton cycles in the complete graph and obtain an infinite sequence of tight explicit constructions.