{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:55:18Z","timestamp":1787324118616,"version":"build-2736575974"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100006115","name":"Institute for Research in Fundamental Sciences","doi-asserted-by":"publisher","award":["99050011"],"award-info":[{"award-number":["99050011"]}],"id":[{"id":"10.13039\/501100006115","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>Haviv [ European J. Combin., 81 (2019), pp. 84--97] has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension over $\\mathbb{R}$. We show that this actually holds for all known topological lower bounds and all fields. We also improve the topological bound he obtained for the minrank parameter over $\\mathbb{R}$---an important graph invariant from coding theory---and show that this bound is actually valid for all fields as well. The notion of independent representation over a matroid is introduced and used in a general theorem having these results as corollaries. Related complexity results are also discussed.<\/jats:p>","DOI":"10.1137\/19m1295921","type":"journal-article","created":{"date-parts":[[2021,1,13]],"date-time":"2021-01-13T13:48:13Z","timestamp":1610545693000},"page":"91-104","source":"Crossref","is-referenced-by-count":1,"title":["Topological Bounds for Graph Representations over Any Field"],"prefix":"10.1137","volume":"35","author":[{"given":"Meysam","family":"Alishahi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fr\u00e9d\u00e9ric","family":"Meunier","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,1,13]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2015.05.010"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.22215"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2017.03.011"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/BF03352998"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(78)90023-7"},{"key":"atypb6","unstructured":"H. R. Daneshpajouh, F. Meunier, and G. Mizrahi,\n                      Colorings of Complements of Line Graphs\n                      , preprint,https:\/\/arxiv.org\/abs\/2003.08255, 2020."},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/BF00969645"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(86)90065-8"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.2307\/1969651"},{"key":"atypb10","unstructured":"M. R. Garey and D. S. Johnson,\n                      Computers and Intractability\n                      , W. H. Freeman, San Francisco, 1979."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.1979.1056027"},{"key":"atypb12","first-page":"39","author":"Haviv I.","year":"2019","journal-title":"Germany"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2019.04.006"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2014.04.008"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"L. Lov\u00e1sz,\n                      Kneser's conjecture, chromatic number, and homotopy\n                      , J. Combin. Theory Ser. A, 25 (1978), pp. 319-324.","DOI":"10.1016\/0097-3165(78)90022-5"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.1979.1055985"},{"key":"atypb17","doi-asserted-by":"crossref","unstructured":"L. Lov\u00e1sz,\n                      Graphs and Geometry\n                      , Amer. Math. Soc. Colloq. Publ. 65, AMS, Providence, RI, 2019.","DOI":"10.1090\/coll\/065"},{"key":"atypb18","unstructured":"L. Lov\u00e1sz and K. Vesztergombi,\n                      Geometric Representations of Graphs\n                      , Paul Erd\u00f6s and His Mathematics, II (Budapest, 1999), Bolyai Soc. Math. Stud. 11, J\u00e1nos Bolyai Math. Soc., Budapest, 2002, pp. 471-498."},{"key":"atypb19","unstructured":"J. Matou\u0161ek,\n                      Using the Borsuk-Ulam Theorem, Lectures on Topological Methods in Combinatorics and Geometry\n                      , Written in cooperation with Anders Bj\u00f6rner and G\u00fcnter M. Ziegler, Universitext, Springer-Verlag, Berlin, 2003."},{"key":"atypb20","first-page":"71","volume":"106","author":"Matou\u0161ek J.","year":"2004","journal-title":"Jahresber. Deutsch. Math.-Verein."},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1007\/BF01261326"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(03)00333-0"},{"key":"atypb23","first-page":"1152","author":"Shanmugam K.","year":"2013","journal-title":"IEEE"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.37236\/2302"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-006-0034-x"},{"key":"atypb26","first-page":"285","author":"Tucker A. W.","year":"1946","journal-title":"ON"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/19M1295921","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:18:57Z","timestamp":1787321937000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/19M1295921"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":26,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/19M1295921"],"URL":"https:\/\/doi.org\/10.1137\/19m1295921","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}