{"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-quantum-chromatic-numbers-of-small","title":"On the Quantum Chromatic Numbers of Small Graphs","arxiv_id":"2311.08194","date":"2023-11-14","proceeding":null,"authors":["Olivier Lalonde"],"abstract":"We make two contributions pertaining to the study of the quantum chromatic numbers of small graphs. Firstly, in an elegant paper, Man\\v{c}inska and Roberson [\\textit{Baltic Journal on Modern Computing}, 4(4), 846-859, 2016] gave an example of a graph $G_{14}$ on 14 vertices with quantum chromatic number 4 and classical chromatic number 5, and conjectured that this is the smallest graph exhibiting a separation between the two parameters. We describe a computer-assisted proof of this conjecture, thereby resolving a longstanding open problem in quantum graph theory. Our second contribution pertains to the study of the rank-$r$ quantum chromatic numbers. While it can now be shown that for every $r$, $\\chi_q$ and $\\chi^{(r)}_q$ are distinct, few small examples of separations between these parameters are known. We give the smallest known example of such a separation in the form of a graph $G_{21}$ on 21 vertices with $\\chi_q(G_{21}) = \\chi^{(2)}_q(G_{21}) = 4$ and $ \\xi(G_{21}) = \\chi^{(1)}_q(G_{21}) = \\chi(G_{21}) = 5$. The previous record was held by a graph $G_{msg}$ on 57 vertices that was first considered in the aforementioned paper of Man\\v{c}inska and Roberson and which satisfies $\\chi_q(G_{msg}) = 3$ and $\\chi^{(1)}_q(G_{msg}) = 4$. In addition, $G_{21}$ provides the first provable separation between the parameters $\\chi^{(1)}_q$ and $\\chi^{(2)}_q$. We believe that our techniques for constructing $G_{21}$ and lower bounding its orthogonal rank could be of independent interest.","url_abs":"https://arxiv.org/abs/2311.08194v1","url_pdf":"https://arxiv.org/pdf/2311.08194v1.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-quantum-chromatic-numbers-of-small","repo_url":"https://github.com/lalondeo/quantumcolorings","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}