{"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-clique-covering-numbers-of-johnson","title":"On the clique covering numbers of Johnson graphs","arxiv_id":"2502.15019","date":"2025-02-20","proceeding":null,"authors":["Søren Fuglede Jørgensen"],"abstract":"We initiate a study of the vertex clique covering numbers of Johnson graphs $J(N, k)$, the smallest numbers of cliques necessary to cover the vertices of those graphs. We prove identities for the values of these numbers when $k \\leq 3$, and $k \\geq N - 3$, and using computational methods, we provide explicit values for a range of small graphs. By drawing on connections to coding theory and combinatorial design theory, we prove various bounds on the clique covering numbers for general Johnson graphs, and we show how constant-weight lexicodes can be utilized to create optimal covers of $J(2k, k)$ when $k$ is a small power of two.","url_abs":"https://arxiv.org/abs/2502.15019v1","url_pdf":"https://arxiv.org/pdf/2502.15019v1.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-clique-covering-numbers-of-johnson","repo_url":"https://github.com/Kvantify/johnson-clique-cover","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"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}