{"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/more-tales-of-hoffman-bounds-for-the-vector","title":"More Tales of Hoffman: bounds for the vector chromatic number of a graph","arxiv_id":"1812.02613","date":"2018-12-06","proceeding":null,"authors":["Pawel Wocjan","Clive Elphick","David Anekstein"],"abstract":"Let $\\chi(G)$ denote the chromatic number of a graph and $\\chi_v(G)$ denote the vector chromatic number. For all graphs $\\chi_v(G) \\le \\chi(G)$ and for some graphs $\\chi_v(G) \\ll \\chi(G)$. Galtman proved that Hoffman's well-known lower bound for $\\chi(G)$ is in fact a lower bound for $\\chi_v(G)$. We prove that two more spectral lower bounds for $\\chi(G)$ are also lower bounds for $\\chi_v(G)$. We then use one of these bounds to derive a new characterization of $\\chi_v(G)$.","url_abs":"http://arxiv.org/abs/1812.02613v3","url_pdf":"http://arxiv.org/pdf/1812.02613v3.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":"more-tales-of-hoffman-bounds-for-the-vector","repo_url":"https://github.com/aneksteind/MoreTalesOfHoffman","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":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}