{"pk":64945,"title":"Ubiquity of locally finite graphs with extensive tree-decompositions","subtitle":null,"abstract":"A graph \\(G\\) is said to be ubiquitous, if every graph \\(\\Gamma\\) that contains arbitrarily many disjoint \\(G\\)-minors automatically contains infinitely many disjoint \\(G\\)-minors. The well-known Ubiquity conjecture of Andreae says that every locally finite graph is ubiquitous. In this paper we show that locally finite graphs admitting a certain type of tree-decomposition, which we call an extensive tree-decomposition, are ubiquitous. In particular this includes all locally finite graphs of finite tree-width, and also all locally finite graphs with finitely many ends, all of which have finite degree. It remains an open question whether every locally finite graph admits an extensive tree-decomposition.\n \nMathematics Subject Classifications: 05C83, 05C63\n \nKeywords: Graph minors, infinite graphs, ubiquity","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"Graph minors"},{"word":"infinite graphs"},{"word":"ubiquity"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/7xg5m02t","frozenauthors":[{"first_name":"Nathan","middle_name":"","last_name":"Bowler","name_suffix":"","institution":"Universität Hamburg, Department of Mathematics, Bundesstraße 55 (Geomatikum), 20146 Hamburg, Germany","department":""},{"first_name":"Christian","middle_name":"","last_name":"Elbracht","name_suffix":"","institution":"Universität Hamburg, Department of Mathematics, Bundesstraße 55 (Geomatikum), 20146 Hamburg, Germany","department":""},{"first_name":"Joshua","middle_name":"","last_name":"Erde","name_suffix":"","institution":"Graz University of Technology, Institute of Discrete Mathematics, Steyrergasse 30, 8010 Graz, Austria","department":""},{"first_name":"J.","middle_name":"Pascal","last_name":"Gollin","name_suffix":"","institution":"FAMNIT, University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia","department":""},{"first_name":"Karl","middle_name":"","last_name":"Heuer","name_suffix":"","institution":"Technical University of Denmark, Department of Applied Mathematics and Computer Science, Richard Petersens Plads, Building 322, 2800 Kongens Lyngby, Denmark","department":""},{"first_name":"Max","middle_name":"","last_name":"Pitz","name_suffix":"","institution":"Universität Hamburg, Department of Mathematics, Bundesstraße 55 (Geomatikum), 20146 Hamburg, Germany","department":""},{"first_name":"Maximilian","middle_name":"","last_name":"Teegen","name_suffix":"","institution":"Universität Hamburg, Department of Mathematics, Bundesstraße 55 (Geomatikum), 20146 Hamburg, Germany","department":""}],"date_submitted":"2024-09-25T19:06:42+03:00","date_accepted":"2024-09-25T19:06:42+03:00","date_published":"2024-09-30T10:00:00+03:00","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64945/galley/49755/download/"}]}