{"pk":65019,"title":"Periodic colorings and orientations in infinite graphs","subtitle":null,"abstract":"We study the existence of periodic colorings and orientations in locally finite graphs. A coloring or orientation of a graph \\(G\\) is periodic if the resulting colored or oriented graph is quasi-transitive, meaning that \\(V(G)\\) has finitely many orbits under the action of the group of automorphisms of \\(G\\) preserving the coloring or the orientation. When such a periodic coloring or orientation of \\(G\\) exists, \\(G\\) itself must be quasi-transitive and it is natural to investigate when quasi-transitive graphs have such periodic colorings or orientations. We provide examples of Cayley graphs with no periodic orientation or non-trivial coloring, and examples of quasi-transitive graphs of treewidth 2 without periodic orientation or proper coloring. On the other hand we show that every quasi-transitive graph \\(G\\) of bounded pathwidth has a periodic proper coloring with \\(\\chi(G)\\) colors and a periodic orientation. We relate these problems with techniques and questions from symbolic dynamics and distributed computing and conclude with a number of open problems.\n \nMathematics Subject Classifications: 05C15, 05C25, 20F65\n \nKeywords: Quasi-transitive graphs, periodic colorings, simple groups","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":"Quasi-transitive graphs"},{"word":"periodic colorings"},{"word":"simple groups"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/749907tj","frozenauthors":[{"first_name":"Tara","middle_name":"","last_name":"Abrishami","name_suffix":"","institution":"Department of Mathematics, University of Hamburg, Hamburg, Germany","department":""},{"first_name":"Louis","middle_name":"","last_name":"Esperet","name_suffix":"","institution":"Laboratoire G-SCOP, Université Grenoble Alpes, CNRS, Grenoble, France","department":""},{"first_name":"Ugo","middle_name":"","last_name":"Giocanti","name_suffix":"","institution":"Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland","department":""},{"first_name":"Matthias","middle_name":"","last_name":"Hamann","name_suffix":"","institution":"Department of Mathematics, University of Hamburg, Hamburg, Germany","department":""},{"first_name":"Paul","middle_name":"","last_name":"Knappe","name_suffix":"","institution":"Department of Mathematics, University of Hamburg, Hamburg, Germany","department":""},{"first_name":"Rögnvaldur","middle_name":"G.","last_name":"Möller","name_suffix":"","institution":"Science Institute, University of Iceland, Reykjavík, Iceland","department":""}],"date_submitted":"2026-02-02T09:11:00Z","date_accepted":"2026-02-02T09:11:00Z","date_published":"2025-12-20T08:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/65019/galley/49829/download/"}]}