{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/computational-methods-for-finding-bi-regular","title":"Computational methods for finding bi-regular cages","arxiv_id":"2411.17351","date":"2024-11-26","proceeding":null,"authors":["Jan Goedgebeur","Jorik Jooken","Tibo Van den Eede"],"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.","url_abs":"https://arxiv.org/abs/2411.17351v1","url_pdf":"https://arxiv.org/pdf/2411.17351v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"computational-methods-for-finding-bi-regular","repo_url":"https://github.com/tiboat/bireggirthgraphs","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}