[{"oa":1,"ec_funded":1,"intvolume":"        18","title":"Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs","_id":"22247","publication_status":"published","oa_version":"Published Version","abstract":[{"lang":"eng","text":"A linearly ordered (LO) k-colouring of a hypergraph is a colouring of its vertices with colours 1, …, k such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO k-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring).\r\nHere, we investigate the complexity of approximating the “linearly ordered chromatic number” of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO 3-colourable, and the case that it is not even LO 4-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opršal, Wrochna, and Živný (2023)."}],"keyword":["Constraint satisfaction problem","hypergraph colouring","promise problem","topological methods"],"das_tickbox":"0","year":"2026","OA_type":"gold","arxiv":1,"supplementarymaterial":"no","license":"https://creativecommons.org/licenses/by/4.0/","publication_identifier":{"issn":["1942-3454"],"eissn":["1942-3462"]},"external_id":{"arxiv":["2312.12981"]},"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"project":[{"call_identifier":"FWF","grant_number":"P31312","_id":"26611F5C-B435-11E9-9278-68D0E5697425","name":"Algorithms for Embeddings and Homotopy Theory"},{"name":"IST-BRIDGE: International postdoctoral program","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","grant_number":"101034413","call_identifier":"H2020"}],"publisher":"Association for Computing Machinery","article_number":"10","status":"public","publication":"ACM Transactions on Computation Theory","date_published":"2026-05-04T00:00:00Z","month":"05","ddc":["500"],"date_updated":"2026-07-06T09:06:29Z","department":[{"_id":"UlWa"}],"has_accepted_license":"1","type":"journal_article","issue":"2","corr_author":"1","article_type":"original","acknowledgement":"This research was supported by the Charles University project PRIMUS/21/SCI/014, by the Ministry of Education, Youth\r\nand Sports of the Czech Republic under the project MSCAfellow5_MUNI (CZ.02.01.01/00/22_010/0003229), and by the\r\nAustrian Science Fund (FWF project P31312-N35). This research was funded by UKRI EP/X024431/1 and by a Clarendon\r\nFund Scholarship. This project has received funding from the European Union’s Horizon 2020 research and innovation\r\nprogramme under the Marie Skłodowska-Curie Grant Agreement No 101034413.\r\n","related_material":{"record":[{"id":"15168","status":"public","relation":"earlier_version"}]},"OA_place":"publisher","researchdata_availability":"no","file_date_updated":"2026-07-06T09:03:02Z","day":"04","date_created":"2026-07-05T22:01:37Z","language":[{"iso":"eng"}],"author":[{"full_name":"Filakovský, Marek","first_name":"Marek","last_name":"Filakovský","id":"3E8AF77E-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Nakajima, Tamio Vesa","first_name":"Tamio Vesa","last_name":"Nakajima"},{"first_name":"Jakub","last_name":"Opršal","id":"ec596741-c539-11ec-b829-c79322a91242","orcid":"0000-0003-1245-3456","full_name":"Opršal, Jakub"},{"first_name":"Gianluca","last_name":"Tasinato","id":"0433290C-AF8F-11E9-A4C7-F729E6697425","full_name":"Tasinato, Gianluca"},{"orcid":"0000-0002-1494-0568","full_name":"Wagner, Uli","first_name":"Uli","last_name":"Wagner","id":"36690CA2-F248-11E8-B48F-1D18A9856A87"}],"PlanS_conform":"1","article_processing_charge":"Yes","file":[{"access_level":"open_access","date_created":"2026-07-06T09:03:02Z","relation":"main_file","success":1,"file_name":"2026_TransactionsGraphics_Filakovsky.pdf","creator":"dernst","checksum":"0399ab94085878fc810084845eabd627","date_updated":"2026-07-06T09:03:02Z","file_id":"22252","file_size":941518,"content_type":"application/pdf"}],"scopus_import":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","doi":"10.1145/3779121","volume":18,"citation":{"ieee":"M. Filakovský, T. V. Nakajima, J. Opršal, G. Tasinato, and U. Wagner, “Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs,” <i>ACM Transactions on Computation Theory</i>, vol. 18, no. 2. Association for Computing Machinery, 2026.","ista":"Filakovský M, Nakajima TV, Opršal J, Tasinato G, Wagner U. 2026. Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs. ACM Transactions on Computation Theory. 18(2), 10.","short":"M. Filakovský, T.V. Nakajima, J. Opršal, G. Tasinato, U. Wagner, ACM Transactions on Computation Theory 18 (2026).","chicago":"Filakovský, Marek, Tamio Vesa Nakajima, Jakub Opršal, Gianluca Tasinato, and Uli Wagner. “Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs.” <i>ACM Transactions on Computation Theory</i>. Association for Computing Machinery, 2026. <a href=\"https://doi.org/10.1145/3779121\">https://doi.org/10.1145/3779121</a>.","apa":"Filakovský, M., Nakajima, T. V., Opršal, J., Tasinato, G., &#38; Wagner, U. (2026). Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs. <i>ACM Transactions on Computation Theory</i>. Association for Computing Machinery. <a href=\"https://doi.org/10.1145/3779121\">https://doi.org/10.1145/3779121</a>","ama":"Filakovský M, Nakajima TV, Opršal J, Tasinato G, Wagner U. Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs. <i>ACM Transactions on Computation Theory</i>. 2026;18(2). doi:<a href=\"https://doi.org/10.1145/3779121\">10.1145/3779121</a>","mla":"Filakovský, Marek, et al. “Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs.” <i>ACM Transactions on Computation Theory</i>, vol. 18, no. 2, 10, Association for Computing Machinery, 2026, doi:<a href=\"https://doi.org/10.1145/3779121\">10.1145/3779121</a>."},"quality_controlled":"1"}]