Small graphs and hypergraphs of given degree and girth

TL;DR

This paper explores the relationship between the smallest regular graphs and hypergraphs with given girth and degree, introducing new constructions and record-breaking examples for both graphs and hypergraphs.

Contribution

It establishes a close link between the cage problem for graphs and hypergraphs, extending Cayley graph ideas to hypergraphs and finding new record hypergraphs and graphs.

Findings

Found new record smallest hypergraphs for various parameters.

Discovered new cubic graphs with girths 23 to 32.

Extended Cayley graph techniques to hypergraph constructions.

Abstract

The search for the smallest possible $d$-regular graph of girth $g$ has a long history, and is usually known as the cage problem. This problem has a natural extension to hypergraphs, where we may ask for the smallest number of vertices in a $d$-regular, $r$-uniform hypergraph of given (Berge) girth $g$. We show that these two problems are in fact very closely linked. By extending the ideas of Cayley graphs to the hypergraph context, we find smallest known hypergraphs for various parameter sets. Because of the close link to the cage problem from graph theory, we are able to use these techniques to find new record smallest cubic graphs of girths 23, 24, 28, 29, 30, 31 and 32.