Computational methods for finding bi-regular cages

TL;DR

This paper develops computational methods to find bi-regular cages, providing exhaustive lists for certain parameters, improving bounds on their minimum order, and generalizing existing theorems to enhance upper bounds.

Contribution

It introduces an exhaustive generation algorithm for bi-regular cages, discovers previously unknown cages, and improves bounds through new constructions and theorem generalizations.

Findings

Generated exhaustive lists for 24 parameter sets.

Improved lower bound for n({4,5};7) from 66 to 69.

Enhanced 49 upper bounds and 73 additional bounds via theorem generalization.

Abstract

An $(\{r,m\};g)$-graph is a (simple, undirected) graph of girth $g\geq3$ with vertices of degrees $r$ and $m$ where $2 \leq r < m$ . Given $r,m,g$, we seek the $(\{r,m\};g)$-graphs of minimum order, called $(\{r,m\};g)$-cages or bi-regular cages, whose order is denoted by $n(\{r,m\};g)$. In this paper, we use computational methods for finding $(\{r,m\};g)$-graphs of small order. Firstly, we present an exhaustive generation algorithm, which leads to $\unicode{x2013}$ previously unknown $\unicode{x2013}$ exhaustive lists of $(\{r,m\};g)$-cages for 24 different triples $(r,m,g)$. This also leads to the improvement of the lower bound of $n(\{4,5\};7)$ from 66 to 69. Secondly, we improve 49 upper bounds of $n(\{r,m\};g)$ based on constructions that start from $r$-regular graphs. Lastly, we generalize a theorem by Aguilar, Araujo-Pardo and Berman [arXiv:2305.03290, 2023], leading to 73 additional improved upper bounds.