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.