Computational methods for finding bi-regular cages

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.