{"pk":64944,"title":"On the diameters of friends-and-strangers graphs","subtitle":null,"abstract":"Given simple graphs \\(X\\) and \\(Y\\) on the same number of vertices, the friends-and-strangers graph \\(\\operatorname{FS}(X, Y)\\) has as its vertices all bijections from \\(V(X)\\) to \\(V(Y)\\), where two bijections are adjacent if and only if they differ on two adjacent elements of \\(V(X)\\) with images adjacent in \\(Y\\). We study the diameters of connected components of friends-and-strangers graphs: the diameter of a component of \\(\\operatorname{FS}(X,Y)\\) corresponds to the largest number of swaps necessary to go from one configuration in the component to another. We show that any component of \\(\\operatorname{FS}(\\operatorname{Path}_n, Y)\\) has \\(O(n^2)\\) diameter and that any component of \\(\\operatorname{FS}(\\operatorname{Cycle}_n, Y)\\) has \\(O(n^4)\\) diameter, improvable to \\(O(n^3)\\) whenever \\(\\operatorname{FS}(\\operatorname{Cycle}_n, Y)\\) is connected. Answering a question raised by Alon, Defant, and Kravitz in the negative, we use an explicit construction to show that there exist \\(n\\)-vertex graphs \\(X\\) and \\(Y\\) such that \\(\\operatorname{FS}(X,Y)\\) has a component with \\(e^{\\Omega(n)}\\) diameter. We conclude with several suggestions for future research.\n \nMathematics Subject Classifications: 05C12, 05C35, 05C38\n \nKeywords: Friends-and-strangers graphs, diameter, extremal combinatorics, lower bounds, paths, cycles, token swapping, interchange process","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":"Friends-and-strangers graphs"},{"word":"diameter"},{"word":"extremal combinatorics"},{"word":"lower bounds"},{"word":"paths"},{"word":"cycles"},{"word":"token swapping"},{"word":"interchange process"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/88m7r56b","frozenauthors":[{"first_name":"Ryan","middle_name":"","last_name":"Jeong","name_suffix":"","institution":"Department of Statistics, Stanford University, Stanford, California, U.S.A.","department":""}],"date_submitted":"2024-09-25T18:54:17+03:00","date_accepted":"2024-09-25T18:54:17+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/64944/galley/49754/download/"}]}