{"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/tighter-bounds-on-the-independence-number-of","title":"Tighter Bounds on the Independence Number of the Birkhoff Graph","arxiv_id":"2007.05841","date":"2020-07-11","proceeding":null,"authors":["Leonardo Nagami Coregliano","Fernando Granha Jeronimo"],"abstract":"The Birkhoff graph $\\mathcal{B}_n$ is the Cayley graph of the symmetric group $S_n$, where two permutations are adjacent if they differ by a single cycle. Our main result is a tighter upper bound on the independence number $\\alpha(\\mathcal{B}_n)$ of $\\mathcal{B}_n$, namely, we show that $\\alpha(\\mathcal{B}_n) \\le O(n!/1.97^n)$ improving on the previous known bound of $\\alpha(\\mathcal{B}_n) \\le O(n!/\\sqrt{2}^{n})$ by [Kane-Lovett-Rao, FOCS 2017]. Our approach combines a higher-order version of their representation theoretic techniques with linear programming. With an explicit construction, we also improve their lower bound on $\\alpha(\\mathcal{B}_n)$ by a factor of $n/2$. This construction is based on a proper coloring of $\\mathcal{B}_n$, which also gives an upper bound on the chromatic number $\\chi(\\mathcal{B}_n)$ of $\\mathcal{B}_n$. Via known connections, the upper bound on $\\alpha(\\mathcal{B}_n)$ implies alphabet size lower bounds for a family of maximally recoverable codes on grid-like topologies.","url_abs":"http://arxiv.org/abs/2007.05841v2","url_pdf":"http://arxiv.org/pdf/2007.05841v2.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":"tighter-bounds-on-the-independence-number-of","repo_url":"https://github.com/lenacore/birkhoff_code","is_official":1,"mentioned_in_paper":1,"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}