CombinatoricaPub Date : 2025-08-07DOI: 10.1007/s00493-025-00169-9
Freddie Illingworth, Richard Lang, Alp Müyesser, Olaf Parczyk, Amedeo Sgueglia
{"title":"Spanning Spheres in Dirac Hypergraphs","authors":"Freddie Illingworth, Richard Lang, Alp Müyesser, Olaf Parczyk, Amedeo Sgueglia","doi":"10.1007/s00493-025-00169-9","DOIUrl":"https://doi.org/10.1007/s00493-025-00169-9","url":null,"abstract":"<p>We show that a <i>k</i>-uniform hypergraph on <i>n</i> vertices has a spanning subgraph homeomorphic to the <span><span style=\"\"></span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.614ex\" role=\"img\" style=\"vertical-align: -0.706ex;\" viewbox=\"0 -821.4 3023.9 1125.3\" width=\"7.023ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMAIN-28\" y=\"0\"></use><use x=\"389\" xlink:href=\"#MJMATHI-6B\" y=\"0\"></use><use x=\"1133\" xlink:href=\"#MJMAIN-2212\" y=\"0\"></use><use x=\"2133\" xlink:href=\"#MJMAIN-31\" y=\"0\"></use><use x=\"2634\" xlink:href=\"#MJMAIN-29\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">(k - 1)</script></span>-dimensional sphere provided that <i>H</i> has no isolated vertices and each set of <span><span style=\"\"></span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.109ex\" role=\"img\" style=\"vertical-align: -0.305ex;\" viewbox=\"0 -777 2244.9 908.2\" width=\"5.214ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-6B\" y=\"0\"></use><use x=\"743\" xlink:href=\"#MJMAIN-2212\" y=\"0\"></use><use x=\"1744\" xlink:href=\"#MJMAIN-31\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">k - 1</script></span> vertices supported by an edge is contained in at least <span><span style=\"\"></span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.609ex\" role=\"img\" style=\"vertical-align: -0.705ex;\" viewbox=\"0 -820.1 4689.4 1123.4\" width=\"10.892ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-6E\" y=\"0\"></use><use x=\"600\" xlink:href=\"#MJMAIN-2F\" y=\"0\"></use><use x=\"1101\" xlink:href=\"#MJMAIN-32\" y=\"0\"></use><use x=\"1823\" xlink:href=\"#MJMAIN-2B\" y=\"0\"></use><use x=\"2824\" xlink:href=\"#MJMATHI-6F\" y=\"0\"></use><use x=\"3309\" xlink:href=\"#MJMAIN-28\" y=\"0\"></use><use x=\"3699\" xlink:href=\"#MJMATHI-6E\" y=\"0\"></use><use x=\"4299\" xlink:href=\"#MJMAIN-29\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">n/2 + o(n)</script></span> edges. This gives a topological extension of Dirac’s theorem and asymptotically confirms a conjecture of Georgakopoulos, Haslegrave, Montgomery, and Narayanan. Unlike typical results in the area, our proof does not rely on the Absorption Method, the Regularity Lemma or the Blow-up Lemma. Instead, we use a recently introduced framework that is based on covering the vertex set of the host graph with a family of complete blow-ups.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"62 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145003440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
CombinatoricaPub Date : 2025-07-04DOI: 10.1007/s00493-025-00160-4
Bertrand Guenin, Cheolwon Heo, Irene Pivotto
{"title":"Signed graphs with the same even cycles","authors":"Bertrand Guenin, Cheolwon Heo, Irene Pivotto","doi":"10.1007/s00493-025-00160-4","DOIUrl":"https://doi.org/10.1007/s00493-025-00160-4","url":null,"abstract":"<p>Whitney proved that if two 3-connected graphs <i>G</i> and <span><span style=\"\"></span><span data-mathml='<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mi>G</mi><mo>&#x2032;</mo></msup></math>' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"2.113ex\" role=\"img\" style=\"vertical-align: -0.205ex;\" viewbox=\"0 -821.4 1081.3 909.7\" width=\"2.511ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-47\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"1112\" xlink:href=\"#MJMAIN-2032\" y=\"513\"></use></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mi>G</mi><mo>′</mo></msup></math></span></span><script type=\"math/tex\">G'</script></span> have the same set of cycles (or equivalently, the same set of cuts) then <span><span style=\"\"></span><span data-mathml='<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi><mo>=</mo><msup><mi>G</mi><mo>&#x2032;</mo></msup></math>' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"2.113ex\" role=\"img\" style=\"vertical-align: -0.205ex;\" viewbox=\"0 -821.4 3201.9 909.7\" width=\"7.437ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-47\" y=\"0\"></use><use x=\"1064\" xlink:href=\"#MJMAIN-3D\" y=\"0\"></use><g transform=\"translate(2120,0)\"><use x=\"0\" xlink:href=\"#MJMATHI-47\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"1112\" xlink:href=\"#MJMAIN-2032\" y=\"513\"></use></g></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi><mo>=</mo><msup><mi>G</mi><mo>′</mo></msup></math></span></span><script type=\"math/tex\">G=G'</script></span>. We characterize when two 4-connected signed graphs have the same set of even cycles, and we characterize when two 4-connected grafts have the same set of even cuts.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"24 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145003495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
CombinatoricaPub Date : 2025-07-04DOI: 10.1007/s00493-025-00165-z
Joseph Paul MacManus
{"title":"Fat Minors in Finitely Presented Groups","authors":"Joseph Paul MacManus","doi":"10.1007/s00493-025-00165-z","DOIUrl":"https://doi.org/10.1007/s00493-025-00165-z","url":null,"abstract":"<p>We show that a finitely presented group virtually admits a planar Cayley graph if and only if it is asymptotically minor-excluded, partially answering a conjecture of Georgakopoulos and Papasoglu in the affirmative.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"31 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145003399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
CombinatoricaPub Date : 2025-06-23DOI: 10.1007/s00493-025-00161-3
António Girão, Shoham Letzter
{"title":"Partitioning a tournament into sub-tournaments of high connectivity","authors":"António Girão, Shoham Letzter","doi":"10.1007/s00493-025-00161-3","DOIUrl":"https://doi.org/10.1007/s00493-025-00161-3","url":null,"abstract":"<p>We prove that there exists a constant <span><span style=\"\"></span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"1.912ex\" role=\"img\" style=\"vertical-align: -0.205ex;\" viewbox=\"0 -735.2 2268.1 823.4\" width=\"5.268ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-63\" y=\"0\"></use><use x=\"711\" xlink:href=\"#MJMAIN-3E\" y=\"0\"></use><use x=\"1767\" xlink:href=\"#MJMAIN-30\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">c > 0</script></span> such that the vertices of every strongly <span><span style=\"\"></span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.013ex\" role=\"img\" style=\"vertical-align: -0.205ex;\" viewbox=\"0 -778.3 2039.4 866.5\" width=\"4.737ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-63\" y=\"0\"></use><use x=\"655\" xlink:href=\"#MJMAIN-22C5\" y=\"0\"></use><use x=\"1156\" xlink:href=\"#MJMATHI-6B\" y=\"0\"></use><use x=\"1677\" xlink:href=\"#MJMATHI-74\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">c cdot kt</script></span>-connected tournament can be partitioned into <i>t</i> parts, each of which induces a strongly <i>k</i>-connected tournament. This is clearly tight up to a constant factor, and it confirms a conjecture of Kühn, Osthus and Townsend (2016).</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"38 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145003400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
CombinatoricaPub Date : 2025-06-23DOI: 10.1007/s00493-025-00162-2
Christos A. Athanasiadis
{"title":"Local h-polynomials, Uniform Triangulations and Real-rootedness","authors":"Christos A. Athanasiadis","doi":"10.1007/s00493-025-00162-2","DOIUrl":"https://doi.org/10.1007/s00493-025-00162-2","url":null,"abstract":"<p>The local <i>h</i>-polynomial was introduced by Stanley as a fundamental enumerative invariant of a triangulation <span><span style=\"\"></span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.013ex\" role=\"img\" style=\"vertical-align: -0.205ex;\" viewbox=\"0 -778.3 833.5 866.5\" width=\"1.936ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMAIN-394\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">Delta</script></span> of a simplex. This polynomial is known to have nonnegative and symmetric coefficients and is conjectured to be <span><span style=\"\"></span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"1.909ex\" role=\"img\" style=\"vertical-align: -0.705ex;\" viewbox=\"0 -518.7 543.5 822.1\" width=\"1.262ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-3B3\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">gamma</script></span>-positive when <span><span style=\"\"></span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.009ex\" role=\"img\" style=\"vertical-align: -0.205ex;\" viewbox=\"0 -777 833.5 865.1\" width=\"1.936ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMAIN-394\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">Delta</script></span> is flag. This paper shows that the local <i>h</i>-polynomial has the stronger property of being real-rooted when <span><span style=\"\"></span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.013ex\" role=\"img\" style=\"vertical-align: -0.205ex;\" viewbox=\"0 -778.3 833.5 866.5\" width=\"1.936ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMAIN-394\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">Delta</script></span> is the barycentric subdivision of an arbitrary geometric triangulation <span><span style=\"\"></span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"1.912ex\" role=\"img\" style=\"vertical-align: -0.205ex;\" viewbox=\"0 -735.2 625.5 823.4\" width=\"1.453ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMAIN-393\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">Gamma</script></span> of the simplex. An analogous result for edgewise subdivisions is proven. The proofs are based on a new combinatorial formula for the local <i>h</i>-polynomial of <span><span style=\"\"></span><span","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"51 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145003411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}