Pub Date : 2025-08-07DOI: 10.1007/s00493-025-00166-y
Peter Frankl, Jian Wang
<p>We consider families, <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 -735.2 829.5 823.4" width="1.927ex" 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="#MJCAL-46" y="0"></use></g></svg></span><script type="math/tex">mathcal {F}</script></span> of <i>k</i>-subsets of an <i>n</i>-set. For integers <span><span style=""></span><span style="font-size: 100%; display: inline-block;" tabindex="0"><svg focusable="false" height="2.213ex" role="img" style="vertical-align: -0.505ex;" viewbox="0 -735.2 2286.1 952.8" width="5.31ex" 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-72" y="0"></use><use x="729" xlink:href="#MJMAIN-2265" y="0"></use><use x="1785" xlink:href="#MJMAIN-32" y="0"></use></g></svg></span><script type="math/tex">rge 2</script></span>, <span><span style=""></span><span style="font-size: 100%; display: inline-block;" tabindex="0"><svg focusable="false" height="2.213ex" role="img" style="vertical-align: -0.505ex;" viewbox="0 -735.2 2196.1 952.8" width="5.101ex" 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-74" y="0"></use><use x="639" xlink:href="#MJMAIN-2265" y="0"></use><use x="1695" xlink:href="#MJMAIN-31" y="0"></use></g></svg></span><script type="math/tex">tge 1</script></span>, <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 -735.2 829.5 823.4" width="1.927ex" 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="#MJCAL-46" y="0"></use></g></svg></span><script type="math/tex">mathcal {F}</script></span> is called <i>r</i>-wise <i>t</i>-intersecting if any <i>r</i> of its members have at least <i>t</i> elements in common. The most natural construction of such a family is the full <i>t</i>-star, consisting of all <i>k</i>-sets containing a fixed <i>t</i>-set. In the case <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 2286.1 823.4" width="5.31ex" 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-72" y="0"></use><use x="729" xlink:href="#MJMAIN-3D" y="0"></use><use x="1785" xlink:href="#MJMAIN-32" y="0"></use></g></svg></span><scri
我们考虑族，mathcal F{ （n集合的k个子集）对于整数r }ge 2, t ge 1, mathcal，如果它的任意r个元素至少有t个相同的元素，则{F}称为r向t相交。这类族最自然的构造是全t星，它由包含一个固定t集的所有k个集合组成。在r=2的情况下，精确Erdős-Ko-Rado定理表明，当n ge (t+1)（k-t+1）时，完整t星最大。在本文中，我们证明了对于n ge (2.5t)^{1/(r-1)}(k-t)+k，当r ge 3时，完整的t星是最大的。示例表明，指数frac{1}{r-1}是最好的可能。这比巴洛格和林茨最近的结果有了相当大的改善。
{"title":"On r-wise t-intersecting Uniform Families","authors":"Peter Frankl, Jian Wang","doi":"10.1007/s00493-025-00166-y","DOIUrl":"https://doi.org/10.1007/s00493-025-00166-y","url":null,"abstract":"<p>We consider families, <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 -735.2 829.5 823.4\" width=\"1.927ex\" 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=\"#MJCAL-46\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">mathcal {F}</script></span> of <i>k</i>-subsets of an <i>n</i>-set. For integers <span><span style=\"\"></span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.213ex\" role=\"img\" style=\"vertical-align: -0.505ex;\" viewbox=\"0 -735.2 2286.1 952.8\" width=\"5.31ex\" 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-72\" y=\"0\"></use><use x=\"729\" xlink:href=\"#MJMAIN-2265\" y=\"0\"></use><use x=\"1785\" xlink:href=\"#MJMAIN-32\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">rge 2</script></span>, <span><span style=\"\"></span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.213ex\" role=\"img\" style=\"vertical-align: -0.505ex;\" viewbox=\"0 -735.2 2196.1 952.8\" width=\"5.101ex\" 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-74\" y=\"0\"></use><use x=\"639\" xlink:href=\"#MJMAIN-2265\" y=\"0\"></use><use x=\"1695\" xlink:href=\"#MJMAIN-31\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">tge 1</script></span>, <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 -735.2 829.5 823.4\" width=\"1.927ex\" 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=\"#MJCAL-46\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">mathcal {F}</script></span> is called <i>r</i>-wise <i>t</i>-intersecting if any <i>r</i> of its members have at least <i>t</i> elements in common. The most natural construction of such a family is the full <i>t</i>-star, consisting of all <i>k</i>-sets containing a fixed <i>t</i>-set. In the case <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 2286.1 823.4\" width=\"5.31ex\" 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-72\" y=\"0\"></use><use x=\"729\" xlink:href=\"#MJMAIN-3D\" y=\"0\"></use><use x=\"1785\" xlink:href=\"#MJMAIN-32\" y=\"0\"></use></g></svg></span><scri","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"50 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145003398","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}
Pub Date : 2025-08-07DOI: 10.1007/s00493-025-00169-9
Freddie Illingworth, Richard Lang, Alp Müyesser, Olaf Parczyk, Amedeo Sgueglia
We show that a k-uniform hypergraph on n vertices has a spanning subgraph homeomorphic to the -dimensional sphere provided that H has no isolated vertices and each set of vertices supported by an edge is contained in at least 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.
{"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}
Pub Date : 2025-07-04DOI: 10.1007/s00493-025-00160-4
Bertrand Guenin, Cheolwon Heo, Irene Pivotto
Whitney proved that if two 3-connected graphs G and have the same set of cycles (or equivalently, the same set of cuts) then . 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.
{"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}
给定一个图G，它的霍尔比rho (G)= max ＿H{subseteq G }frac{|V(H)|}{alpha (H)}rho (G)= max ＿H{subseteq G }frac{|V(H)|}{alpha (H)}形成了它的分数色数chi ＿f(G) chi ＿f(G)的自然下界。最近的一项研究研究了chi ＿f(G) chi ＿f(G)是否可以用rho (G) rho (G)的（线性）函数有界的问题。Dvořák， Ossona de Mendez和Wu [6， Combinatorica， 2020]通过证明具有有界霍尔比和任意大分数色数的图的存在性给出了否定的答案。在本文中，我们解决了Dvořák等人提出的两个后续问题。第一个问题涉及确定g(n)的增长，定义为所有n顶点图中的最大比率frac{chi _f(G)}{rho (G)}frac{chi _f(G)}{rho (G)}。Dvořák等人得到了Omega (loglog n) le g(n) le O(log n) Omega (loglog n) le g(n) le O（log n）。我们证明了真实值接近上界：g(n)=(log n)^1{-o(1)}g(n)=(log n)^1{-o(1)}。Dvořák等人提出的第二个问题要求存在有界霍尔比的图，任意大的分数色数，使得每个子图都包含一个独立的集合，触及其边缘的恒定分数。我们证明这样的图确实存在。
Pub Date : 2025-07-04DOI: 10.1007/s00493-025-00165-z
Joseph Paul MacManus
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.
{"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}
Pub Date : 2025-06-23DOI: 10.1007/s00493-025-00161-3
António Girão, Shoham Letzter
We prove that there exists a constant such that the vertices of every strongly -connected tournament can be partitioned into t parts, each of which induces a strongly k-connected tournament. This is clearly tight up to a constant factor, and it confirms a conjecture of Kühn, Osthus and Townsend (2016).
{"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}
Pub Date : 2025-06-23DOI: 10.1007/s00493-025-00162-2
Christos A. Athanasiadis
<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
{"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}
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