{"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-the-regularity-planarity-and-edge-bounds","title":"On the Regularity, Planarity and Edge Bounds of Link-irregular Graphs","arxiv_id":"2503.21916","date":"2025-03-27","proceeding":null,"authors":["Alexander Bastien","Omid Khormali"],"abstract":"A graph $G$ is a link-irregular graph if every two distinct vertices of $G$ have non-isomorphic links. The link of a vertex $v$ in $G$ is the subgraph induced by the neighbors of $v$ in $G$. Ali, Chartrand and Zhang [Discussiones Mathematicae. Graph Theory, 45(1) (2025) p.95] conjectured that there exists no regular link-irregular graph. In this paper, we show that the existence of an $r$-regular link irregular graph is very likely for large enough $r$. In particular, we provide a 7-regular link irregular graph on 12 vertices, which serves as a counterexample to the conjecture. Additionally, we prove that no bipartite link-irregular graphs exist, and there are no regular link-irregular graphs on $n$-vertices for $n \\leq 9$. Also, we determine upper and lower bounds for the number of edges of link-irregular graphs. Furthermore, we show the minimum number of edges in a link-irregular graph on the $n$ vertices is $\\Omega(n\\sqrt{\\log n})$. Finally, we prove that all but finitely many link-irregular graphs are non-planar, and there is no regular link-irregular planar graphs.","url_abs":"https://arxiv.org/abs/2503.21916v2","url_pdf":"https://arxiv.org/pdf/2503.21916v2.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-the-regularity-planarity-and-edge-bounds","repo_url":"https://github.com/omidkhormali/link-irregular-graphs","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}