[{"oa":1,"publication_status":"published","publication":"Discrete Mathematics","OA_place":"publisher","language":[{"iso":"eng"}],"acknowledgement":"We thank an anonymous referee for pointing out an error in an earlier version of Theorem 3.1. We also thank an anonymous referee for pointing out numerous typos in an earlier version of the paper.","title":"Subchromatic numbers of powers of graphs with excluded minors","date_published":"2025-04-01T00:00:00Z","abstract":[{"lang":"eng","text":"A k-subcolouring of a graph G is a function f : V (G) → {0,...,k − 1} such that the set of\r\nvertices coloured i induce a disjoint union of cliques. The subchromatic number, χsub(G),\r\nis the minimum k such that G admits a k-subcolouring. Nešetril, ˇ Ossona de Mendez,\r\nPilipczuk, and Zhu (2020), recently raised the problem of finding tight upper bounds for\r\nχsub(G2) when G is planar. We show that χsub(G2) ≤ 43 when G is planar, improving\r\ntheir bound of 135. We give even better bounds when the planar graph G has larger girth.\r\nMoreover, we show that χsub(G3) ≤ 95, improving the previous bound of 364. For these\r\nwe adapt some recent techniques of Almulhim and Kierstead (2022), while also extending\r\nthe decompositions of triangulated planar graphs of Van den Heuvel, Ossona de Mendez,\r\nQuiroz, Rabinovich and Siebertz (2017), to planar graphs of arbitrary girth. Note that these\r\ndecompositions are the precursors of the graph product structure theorem of planar graphs.\r\nWe give improved bounds for χsub(Gp) for all p ≥ 2, whenever G has bounded treewidth,\r\nbounded simple treewidth, bounded genus, or excludes a clique or biclique as a minor.\r\nFor this we introduce a family of parameters which form a gradation between the strong\r\nand the weak colouring numbers. We give upper bounds for these parameters for graphs\r\ncoming from such classes.\r\nFinally, we give a 2-approximation algorithm for the subchromatic number of graphs\r\nhaving a layering in which each layer has bounded cliquewidth and this layering is\r\ncomputable in polynomial time (like the class of all dth powers of planar graphs, for fixed\r\nd). This algorithm works even if the power p and the graph G is unknown."}],"publication_identifier":{"issn":["0012-365X"]},"doi":"10.1016/j.disc.2024.114377","date_updated":"2025-09-30T10:25:15Z","issue":"4","corr_author":"1","intvolume":"       348","article_processing_charge":"Yes (via OA deal)","arxiv":1,"_id":"19002","ddc":["510"],"date_created":"2025-02-05T06:51:08Z","type":"journal_article","citation":{"apa":"Cortés, P. P., Kumar, P., Moore, B., Ossona de Mendez, P., &#38; Quiroz, D. A. (2025). Subchromatic numbers of powers of graphs with excluded minors. <i>Discrete Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.disc.2024.114377\">https://doi.org/10.1016/j.disc.2024.114377</a>","ama":"Cortés PP, Kumar P, Moore B, Ossona de Mendez P, Quiroz DA. Subchromatic numbers of powers of graphs with excluded minors. <i>Discrete Mathematics</i>. 2025;348(4). doi:<a href=\"https://doi.org/10.1016/j.disc.2024.114377\">10.1016/j.disc.2024.114377</a>","ieee":"P. P. Cortés, P. Kumar, B. Moore, P. Ossona de Mendez, and D. A. Quiroz, “Subchromatic numbers of powers of graphs with excluded minors,” <i>Discrete Mathematics</i>, vol. 348, no. 4. Elsevier, 2025.","chicago":"Cortés, Pedro P., Pankaj Kumar, Benjamin Moore, Patrice Ossona de Mendez, and Daniel A. Quiroz. “Subchromatic Numbers of Powers of Graphs with Excluded Minors.” <i>Discrete Mathematics</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.disc.2024.114377\">https://doi.org/10.1016/j.disc.2024.114377</a>.","ista":"Cortés PP, Kumar P, Moore B, Ossona de Mendez P, Quiroz DA. 2025. Subchromatic numbers of powers of graphs with excluded minors. Discrete Mathematics. 348(4), 114377.","mla":"Cortés, Pedro P., et al. “Subchromatic Numbers of Powers of Graphs with Excluded Minors.” <i>Discrete Mathematics</i>, vol. 348, no. 4, 114377, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.disc.2024.114377\">10.1016/j.disc.2024.114377</a>.","short":"P.P. Cortés, P. Kumar, B. Moore, P. Ossona de Mendez, D.A. Quiroz, Discrete Mathematics 348 (2025)."},"month":"04","department":[{"_id":"MaKw"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","external_id":{"isi":["001401656900001"],"arxiv":["2306.02195"]},"volume":348,"article_type":"original","day":"01","status":"public","scopus_import":"1","publisher":"Elsevier","file":[{"relation":"main_file","success":1,"content_type":"application/pdf","date_created":"2025-05-05T12:56:12Z","date_updated":"2025-05-05T12:56:12Z","creator":"dernst","file_name":"2025_DiscreteMath_Cortes.pdf","checksum":"6723cbb02b6aea0d05f37d167da00c03","file_id":"19657","file_size":850988,"access_level":"open_access"}],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"has_accepted_license":"1","OA_type":"hybrid","year":"2025","oa_version":"Published Version","article_number":"114377","file_date_updated":"2025-05-05T12:56:12Z","quality_controlled":"1","author":[{"last_name":"Cortés","first_name":"Pedro P.","full_name":"Cortés, Pedro P."},{"full_name":"Kumar, Pankaj","first_name":"Pankaj","last_name":"Kumar"},{"id":"6dc1a1be-bf1c-11ed-8d2b-d044840f49d6","full_name":"Moore, Benjamin","first_name":"Benjamin","last_name":"Moore"},{"first_name":"Patrice","last_name":"Ossona de Mendez","full_name":"Ossona de Mendez, Patrice"},{"full_name":"Quiroz, Daniel A.","first_name":"Daniel A.","last_name":"Quiroz"}],"isi":1}]