{"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/exhaustive-generation-of-edge-girth-regular","title":"Exhaustive generation of edge-girth-regular graphs","arxiv_id":"2401.08271","date":"2024-01-16","proceeding":null,"authors":["Jan Goedgebeur","Jorik Jooken"],"abstract":"Edge-girth-regular graphs (abbreviated as $egr$ graphs) are a class of highly regular graphs. More specifically, for integers $v$, $k$, $g$ and $\\lambda$ an $egr(v,k,g,\\lambda)$ graph is a $k$-regular graph with girth $g$ on $v$ vertices such that every edge is contained in exactly $\\lambda$ cycles of length $g$. The central problem in this paper is determining $n(k,g,\\lambda)$, which is defined as the smallest integer $v$ such that an $egr(v,k,g,\\lambda)$ graph exists (or $\\infty$ if no such graph exists) as well as determining the corresponding extremal graphs. We propose a linear time algorithm for computing how often an edge is contained in a cycle of length $g$, given a graph with girth $g$. We use this as one of the building blocks to propose another algorithm that can exhaustively generate all $egr(v,k,g,\\lambda)$ graphs for fixed parameters $v, k, g$ and $\\lambda$. We implement this algorithm and use it in a large-scale computation to obtain several new extremal graphs and improvements for lower and upper bounds from the literature for $n(k,g,\\lambda)$. Among others, we show that $n(3,6,2)=24, n(3,8,8)=40, n(3,9,6)=60, n(3,9,8)=60, n(4,5,1)=30, n(4,6,9)=35, n(6,5,20)=42$ and we disprove a conjecture made by Araujo-Pardo and Leemans [Discrete Math. 345(10):112991 (2022)] for the cubic girth 8 and girth 12 cases. Based on our computations, we conjecture that $n(3,7,6)=n(3,8,10)=n(3,8,12)=n(3,8,14)=\\infty.$","url_abs":"https://arxiv.org/abs/2401.08271v2","url_pdf":"https://arxiv.org/pdf/2401.08271v2.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":"exhaustive-generation-of-edge-girth-regular","repo_url":"https://github.com/jorikjooken/edgegirthregulargraphs","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2401.08271","atlas_url":"https://app.syntology.ai/?focus=2401.08271","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}