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.
@article{10_37236_11765,
author = {Grahame Erskine and James Tuite},
title = {Small graphs and hypergraphs of given degree and girth},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {1},
doi = {10.37236/11765},
zbl = {1511.05094},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11765/}
}
TY - JOUR
AU - Grahame Erskine
AU - James Tuite
TI - Small graphs and hypergraphs of given degree and girth
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/11765/
DO - 10.37236/11765
ID - 10_37236_11765
ER -
%0 Journal Article
%A Grahame Erskine
%A James Tuite
%T Small graphs and hypergraphs of given degree and girth
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/11765/
%R 10.37236/11765
%F 10_37236_11765
Grahame Erskine; James Tuite. Small graphs and hypergraphs of given degree and girth. The electronic journal of combinatorics, Tome 30 (2023) no. 1. doi: 10.37236/11765
