The maximum diameter of $d$-dimensional simplicial complexes

TL;DR

This paper determines the maximum diameter of strongly connected d-dimensional simplicial complexes on n vertices for large n, settling a 2013 problem and characterizing Euler tours in hypergraphs.

Contribution

It provides the exact maximum diameter for fixed d and large n, improving previous bounds and resolving Santos's longstanding question.

Findings

Maximum diameter determined for large n and fixed d

Characterization of Euler tours in complete hypergraphs

Improved bounds over previous results

Abstract

For every fixed dimension $d$ and sufficiently large $n$, we determine the maximum possible diameter of a strongly connected $d$-dimensional simplicial complex on $n$ vertices. This improves on a sequence of previous results and settles a problem of Santos from 2013. On the way, as a special case, we also characterise the existence of an extra-tight Euler tour in the complete $d$-uniform hypergraph on $n$ vertices.