The maximum diameter of 2-dimensional simplicial complexes

TL;DR

This paper determines the exact maximum diameter of 2-dimensional simplicial complexes on n vertices and explores related combinatorial problems involving Hamilton cycles in complete graphs.

Contribution

It provides an explicit construction for the maximum diameter of 2D simplicial complexes and introduces new tight constructions for Hamilton cycle packings.

Findings

Exact maximum diameter for 2D complexes on n vertices

Explicit constructions for Hamilton cycle packings

Open problem on square packings in complete graphs

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.