[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84757-en":3,"doc-seo-84757-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84757,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","Hamilton Paths and Cycles in Flip Graphs of (Almost-)Perfect Matchings","Hamilton paths and cycles are studied in flip graphs defined on matchings of complete graphs or complete bipartite graphs, with flips given as local edge exchanges. Perfect matchings (even number of vertices) and almost-perfect matchings (odd number of vertices, with one unmatched vertex) form the vertex sets. Two perfect matchings are adjacent when they differ by a 2-flip (a 4-cycle), while almost-perfect matchings use 1-flips (an edge exchange). Combinatorial results establish Hamilton-connectedness and Hamilton-laceability for bipartite cases, and address a prior open problem. Geometric non-crossing embeddings yield exponential path/cycle exclusions, with a near-spanning cycle constructed in convex position.","arXiv :2607 .04687v2 [math .CO] 7 Jul 2026  \nHAMILTON PATHS AND CYCLES IN FLIP GRAPHS OF  \n(ALMOST-)PERFECT MATCHINGS  \nSOFIA BRENNER, JUSTIN DALLANT, LINDA KLEIST, ROBERT LAUFF, TORSTEN MÜTZE, AND TORBEN SCHÜRENBERG  \nAbstract. We consider the set of matchings of a graph and a local change operation, called a flip, between them. In the combinatorial setting, the base graphs are either complete graphs or complete bipartite graphs, and in the geometric setting, the graphs are embedded on point sets in the plane, with the requirement that edges must be drawn as straight lines and must not cross. For base graphs with an even number of vertices, we consider perfect matchings, i.e., all vertices are matched, and for base graphs with an odd number of vertices, we consider almostperfect matchings, i.e., all but one vertex of the graph are matched. A 2-flip between two perfect matchings exchanges two edges, and a 1-flip between two almost-perfect matchings exchanges one edge. The corresponding flip graph has the set of perfect or almost-perfect matchings as vertices, with pairs of them connected by an edge if and only if they differ in a 2-flip or 1-flip, respectively. In this work, we provide a comprehensive picture of Hamiltonicity properties of these flip graphs, i.e., under which conditions the flip graphs admit spanning paths and cycles.  \nWe prove that the flip graphs in the combinatorial setting are Hamilton-connected, i.e., they admit a Hamilton path between any two vertices, or, if the flip graphs are bipartite, we prove that they are Hamilton-laceable, i.e., they admit a Hamilton path between any two vertices from different partition classes. For almost-perfect matchings in the complete graph under 1-flips this answers a problem raised by Aichholzer, Dorfer, Rieck and Verciani.  \nIn the geometric setting, while the flip graphs are connected, we prove that any path in them misses exponentially many vertices, in particular, they have no Hamilton paths or cycles. For points in convex position and almost-perfect matchings under 1-flips, we complement this by constructing a cycle in the flip graph that visits almost all vertices.  \nWe also introduce a new directed variant of the combinatorial setting, where each matching edge is directed. This gives rise to different types of 2-flips and 1-flips, and leads to a more fine-grained picture of properties of the associated flip graphs, such as number of components, bipartiteness, diameter, Hamilton-connectedness/-laceability.  \n(Sofia Brenner) Fakultät für Mathematik und Informatik, Universität Leipzig, Germany  \n(Justin Dallant) Fakultät Informatik, TU Dresden, Germany (Linda Kleist) Fachbereich Informatik, Universität Hamburg, Germany (Robert Lauff) Institut für Mathematik, TU Berlin, Germany  \n(Torsten Mütze) Institut für Mathematik, Universität Kassel, Germany  \n(Torben Schürenberg) Zentrum für Industriemathematik, Universität Bremen, Germany  \nE-mail addresses: [sofia.brenner@uni-leipzig.de](sofia.brenner@uni-leipzig.de) , justin.dallant@tu-dresden.de, [linda.kleist@uni-hamburg.de](linda.kleist@uni-hamburg.de) , [lauff@math.tu-berlin.de](lauff@math.tu-berlin.de) , [tmuetze@mathematik.uni-kassel.de](tmuetze@mathematik.uni-kassel.de) ,  \n[torsch@uni-bremen.de](torsch@uni-bremen.de).  \nKey words and phrases . Matching, flip graph, Hamilton cycle, complete graph, non-crossing, perfect, almostperfect.  \nThis project was supported by German Science Foundation grant 522790373 .  \n2 HAMILTON PATHS AND CYCLES IN FLIP GRAPHS OF (ALMOST-)PERFECT MATCHINGS  \n1. Introduction  \nMatchings in graphs are fundamental combinatorial objects, and a large body of classical results and algorithms in graph theory and optimization is devoted to them. Think for example of Hall’s marriage theorem, Tutte’s 1-factor theorem, Petersen’s theorem, the Hungarian algorithm, Edmonds’ blossom algorithm, the Gale-Shapley algorithm, and the variety of min-max theorems that link matchings to other graph objects.  \nThe","cbCaipZ2U5J8bDwu","https://ap.wps.com/l/cbCaipZ2U5J8bDwu","pdf",3328947,1,44,"English","en",105,"# Introduction\n## Matchings and flip operations\n## Hamiltonicity objectives and prior work\n## Combinatorial results\n## Geometric results\n## Directed flip-graph variant","[{\"question\":\"What are flip graphs in this work, and how are their edges defined?\",\"answer\":\"Vertices are perfect matchings or almost-perfect matchings of a base graph. Two vertices are connected when the corresponding matchings differ by a local flip: a 2-flip for perfect matchings and a 1-flip for almost-perfect matchings.\"},{\"question\":\"What Hamiltonicity properties are proven in the combinatorial setting?\",\"answer\":\"The flip graphs are Hamilton-connected in the combinatorial case, and in bipartite instances they are Hamilton-laceable, guaranteeing Hamilton paths between appropriate pairs of vertices.\"},{\"question\":\"How do non-crossing geometric embeddings affect Hamilton paths and cycles?\",\"answer\":\"Although geometric flip graphs are connected, any path omits exponentially many vertices, implying the absence of Hamilton paths or Hamilton cycles. For points in convex position with almost-perfect matchings under 1-flips, the authors also construct a cycle visiting almost all vertices.\"}]",1784198074,111,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"hamilton-paths-and-cycles-in-flip-graphs-of-almost-perfect-matchings","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/hamilton-paths-and-cycles-in-flip-graphs-of-almost-perfect-matchings/84757/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What are flip graphs in this work, and how are their edges defined?","Question",{"text":75,"@type":76},"Vertices are perfect matchings or almost-perfect matchings of a base graph. Two vertices are connected when the corresponding matchings differ by a local flip: a 2-flip for perfect matchings and a 1-flip for almost-perfect matchings.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What Hamiltonicity properties are proven in the combinatorial setting?",{"text":80,"@type":76},"The flip graphs are Hamilton-connected in the combinatorial case, and in bipartite instances they are Hamilton-laceable, guaranteeing Hamilton paths between appropriate pairs of vertices.",{"name":82,"@type":73,"acceptedAnswer":83},"How do non-crossing geometric embeddings affect Hamilton paths and cycles?",{"text":84,"@type":76},"Although geometric flip graphs are connected, any path omits exponentially many vertices, implying the absence of Hamilton paths or Hamilton cycles. For points in convex position with almost-perfect matchings under 1-flips, the authors also construct a cycle visiting almost all vertices.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]