{"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/on-k-g-graphs-without-g-1-cycles","title":"On $(k,g)$-Graphs without $(g+1)$-Cycles","arxiv_id":"2411.19023","date":"2024-11-28","proceeding":null,"authors":["Leonard Chidiebere Eze","Robert Jajcay","Jorik Jooken"],"abstract":"A $(k,g,\\underline{g+1})$-graph is a $k$-regular graph of girth $g$ which does not contain cycles of length $g+1$. Such graphs are known to exist for all parameter pairs $k \\geq 3, g \\geq 3 $, and we focus on determining the orders $n(k,g,\\underline{g+1})$ of the smallest $(k,g,\\underline{g+1})$-graphs. This problem can be viewed as a special case of the previously studied Girth Pair Problem, the problem of finding the order of a smallest $k$-regular graph in which the length of a smallest even length cycle and the length of a smallest odd length cycle are prescribed. When considering the case of an odd girth $g$, this problem also yields results towards the Cage Problem, the problem of finding the order of a smallest $k$-regular graph of girth $g$. We establish the monotonicity of the function $n(k,g,\\underline{g+1})$ with respect to increasing $g$, and present universal lower bounds for the values $n(k,g,\\underline{g+1})$. We propose an algorithm for generating all $(k,g,\\underline{g+1})$-graphs on $n$ vertices, use this algorithm to determine several of the smaller values $n(k,g,\\underline{g+1})$, and discuss various approaches to finding smallest $(k,g,\\underline{g+1})$-graphs within several classes of highly symmetrical graphs.","url_abs":"https://arxiv.org/abs/2411.19023v1","url_pdf":"https://arxiv.org/pdf/2411.19023v1.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":"on-k-g-graphs-without-g-1-cycles","repo_url":"https://github.com/JorikJooken/KGNoGPlus1Graphs","is_official":1,"mentioned_in_paper":0,"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}