European Journal of Combinatorics最新文献

{"title":"The Frank number and nowhere-zero flows on graphs","authors":"Jan Goedgebeur ,&nbsp;Edita Máčajová ,&nbsp;Jarne Renders","doi":"10.1016/j.ejc.2025.104127","DOIUrl":"10.1016/j.ejc.2025.104127","url":null,"abstract":"<div><div>An edge <span><math><mi>e</mi></math></span> of a graph <span><math><mi>G</mi></math></span> is called <em>deletable</em> for some orientation <span><math><mi>o</mi></math></span> if the restriction of <span><math><mi>o</mi></math></span> to <span><math><mrow><mi>G</mi><mo>−</mo><mi>e</mi></mrow></math></span> is a strong orientation. Inspired by a problem of Frank, in 2021 Hörsch and Szigeti proposed a new parameter for 3-edge-connected graphs, called the Frank number, which refines <span><math><mi>k</mi></math></span>-edge-connectivity. The <em>Frank number</em> is defined as the minimum number of orientations of <span><math><mi>G</mi></math></span> for which every edge of <span><math><mi>G</mi></math></span> is deletable in at least one of them. They showed that every 3-edge-connected graph has Frank number at most 7 and that in case these graphs are also 5-edge-colourable the parameter is at most 3. Here we strengthen both results by showing that every 3-edge-connected graph has Frank number at most 4 and that every graph which is 3-edge-connected and 3-edge-colourable has Frank number 2. The latter also confirms a conjecture by Barát and Blázsik. Furthermore, we prove two sufficient conditions for cubic graphs to have Frank number 2 and use them in an algorithm to computationally show that the Petersen graph is the only cyclically 4-edge-connected cubic graph up to 36 vertices having Frank number greater than 2.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104127"},"PeriodicalIF":1.0,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Symmetries of voltage operations on polytopes, maps and maniplexes","authors":"Isabel Hubard ,&nbsp;Elías Mochán ,&nbsp;Antonio Montero","doi":"10.1016/j.ejc.2025.104132","DOIUrl":"10.1016/j.ejc.2025.104132","url":null,"abstract":"<div><div>Voltage operations extend traditional geometric and combinatorial operations (such as medial, truncation, prism, and pyramid over a polytope) to operations on maniplexes, maps, polytopes, and hypertopes. In classical operations, the symmetries of the original object remain in the resulting one, but sometimes additional symmetries are created; the same situation arises with voltage operations. We characterise the automorphisms of the new object that are derived from the original one and use this to bound the number of flag orbits (under the action of its automorphism group) of the new object in terms of the original one. The conditions under which the automorphism group of the original object is the same as the automorphism group of the resulting object are given. We also look at the cases where there is additional symmetry which can be accurately described due to the symmetries of the operation itself.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104132"},"PeriodicalIF":1.0,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143360670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On combinatorics of string polytopes in types B and C","authors":"Yunhyung Cho ,&nbsp;Naoki Fujita ,&nbsp;Eunjeong Lee","doi":"10.1016/j.ejc.2025.104126","DOIUrl":"10.1016/j.ejc.2025.104126","url":null,"abstract":"<div><div>A string polytope is a rational convex polytope whose lattice points parametrize a highest weight crystal basis, which is obtained from a string cone by explicit affine inequalities depending on a highest weight. It also inherits geometric information of a flag variety such as toric degenerations, Newton–Okounkov bodies, mirror symmetry, Schubert calculus, and so on. In this paper, we study combinatorial properties of string polytopes in types <span><math><mi>B</mi></math></span> and <span><math><mi>C</mi></math></span> by giving an explicit description of string cones in these types which is analogous to Gleizer–Postnikov’s description of string cones in type <span><math><mi>A</mi></math></span>. As an application, we characterize string polytopes in type <span><math><mi>C</mi></math></span> which are unimodularly equivalent to the Gelfand–Tsetlin polytope in type <span><math><mi>C</mi></math></span> for a specific highest weight.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104126"},"PeriodicalIF":1.0,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143348629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"High-dimensional expanders from Kac–Moody–Steinberg groups","authors":"Laura Grave de Peralta ,&nbsp;Inga Valentiner-Branth","doi":"10.1016/j.ejc.2025.104131","DOIUrl":"10.1016/j.ejc.2025.104131","url":null,"abstract":"<div><div>High-dimensional expanders are a generalization of the notion of expander graphs to simplicial complexes and give rise to a variety of applications in computer science and other fields. We provide a general tool to construct families of bounded degree high-dimensional spectral expanders. Inspired by the work of Kaufman and Oppenheim, we use coset complexes over quotients of Kac–Moody–Steinberg groups of rank <span><math><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></math></span>, <span><math><mi>d</mi></math></span>-spherical and purely <span><math><mi>d</mi></math></span>-spherical. We prove that infinite families of such quotients exist provided that the underlying field is of size at least 4 and the Kac–Moody–Steinberg group is 2-spherical, giving rise to new families of bounded degree high-dimensional expanders. In case the generalized Cartan matrix we consider is affine, we recover the construction of O’Donnell and Pratt from 2022 (and thus also the one by Kaufman and Oppenheim) by considering Chevalley groups as quotients of affine Kac–Moody–Steinberg groups. Moreover, our construction applies to the case where the root system is of type <span><math><msub><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msub></math></span>, a case that was not covered in earlier works.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104131"},"PeriodicalIF":1.0,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143288948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Decomposition horizons and a characterization of stable hereditary classes of graphs","authors":"Samuel Braunfeld ,&nbsp;Jaroslav Nešetřil ,&nbsp;Patrice Ossona de Mendez ,&nbsp;Sebastian Siebertz","doi":"10.1016/j.ejc.2025.104130","DOIUrl":"10.1016/j.ejc.2025.104130","url":null,"abstract":"<div><div>The notions of bounded-size and quasibounded-size decompositions with bounded treedepth base classes are central to the structural theory of graph sparsity introduced by two of the authors years ago, and provide a characterization of both classes with bounded expansions and nowhere dense classes.</div><div>Strong connections of this theory with model theory led to considering first-order transductions, which are logically defined graph transformations, and to initiate a comparative study of combinatorial and model theoretical properties of graph classes, with an emphasis on the model theoretical notions of dependence (or NIP) and stability.</div><div>In this paper, we first prove that the model theoretic notions of dependence and stability are, for hereditary classes of graphs, compatible with quasibounded-size decompositions, in the following sense: every hereditary class with quasibounded-size decompositions with dependent (resp. stable) base classes is itself dependent (resp. stable). This result is obtained in a more general study of “decomposition horizons”, which are class properties compatible with quasibounded-size decompositions.</div><div>We deduce that hereditary classes with quasibounded-size decompositions with bounded shrubdepth base classes are stable. In the second part of the paper, we prove the converse. Thus, we characterize stable hereditary classes of graphs as those hereditary classes that admit quasibounded-size decompositions with bounded shrubdepth base classes. This result is obtained by proving that every hereditary stable class of graphs admits almost nowhere dense quasi-bush representations, thus answering positively a conjecture of Dreier et al.</div><div>These results have several consequences. For example, we show that every graph <span><math><mi>G</mi></math></span> in a stable, hereditary class of graphs <span><math><mi>C</mi></math></span> has a clique or a stable set of size <span><math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mi>C</mi><mo>,</mo><mi>ɛ</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo>−</mo><mi>ɛ</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, for every <span><math><mrow><mi>ɛ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>, which is tight in the sense that it cannot be improved to <span><math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mi>C</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"129 ","pages":"Article 104130"},"PeriodicalIF":1.0,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144588406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Difference ascent sequences and related combinatorial structures","authors":"Yongchun Zang ,&nbsp;Robin D.P. Zhou","doi":"10.1016/j.ejc.2025.104128","DOIUrl":"10.1016/j.ejc.2025.104128","url":null,"abstract":"<div><div>Ascent sequences were introduced by Bousquet-Mélou, Claesson, Dukes and Kitaev, and are in bijection with unlabeled <span><math><mrow><mo>(</mo><mn>2</mn><mo>+</mo><mn>2</mn><mo>)</mo></mrow></math></span>-free posets, Fishburn matrices, permutations avoiding a bivincular pattern of length 3, and Stoimenow matchings. Analogous results for weak ascent sequences have been obtained by Bényi, Claesson and Dukes. Recently, Dukes and Sagan introduced a more general class of sequences which are called <span><math><mi>d</mi></math></span>-ascent sequences. They showed that some maps from the weak case can be extended to bijections for general <span><math><mi>d</mi></math></span> while the extensions of others continue to be injective but not surjective. The main objective of this paper is to restore these injections to bijections. To be specific, we introduce a class of permutations which we call difference <span><math><mi>d</mi></math></span> permutations and a class of factorial posets which we call difference <span><math><mi>d</mi></math></span> posets, both of which are shown to be in bijection with <span><math><mi>d</mi></math></span>-ascent sequences. Moreover, we also give a direct bijection between a class of matrices with a certain column restriction and Fishburn matrices. Our results give answers to several questions posed by Dukes and Sagan.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104128"},"PeriodicalIF":1.0,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143288949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The asymptotic repetition threshold of sequences rich in palindromes","authors":"L’ubomíra Dvořáková ,&nbsp;Karel Klouda ,&nbsp;Edita Pelantová","doi":"10.1016/j.ejc.2025.104124","DOIUrl":"10.1016/j.ejc.2025.104124","url":null,"abstract":"<div><div>The asymptotic critical exponent measures for a sequence the maximum repetition rate of factors of growing length. The infimum of asymptotic critical exponents of sequences of a certain class is called the asymptotic repetition threshold of that class. On the one hand, if we consider the class of all <span><math><mi>d</mi></math></span>-ary sequences with <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, then the asymptotic repetition threshold is equal to one, independently of the alphabet size. On the other hand, for the class of episturmian sequences, the repetition threshold depends on the alphabet size. We focus on rich sequences, i.e., sequences whose factors contain the maximum possible number of distinct palindromes. The class of episturmian sequences forms a subclass of rich sequences. We prove that the asymptotic repetition threshold for the class of rich recurrent <span><math><mi>d</mi></math></span>-ary sequences, with <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, is equal to two, independently of the alphabet size.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104124"},"PeriodicalIF":1.0,"publicationDate":"2025-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176373","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-existence of two infinite families of strongly regular graphs","authors":"Jack H. Koolen ,&nbsp;Brhane Gebremichel ,&nbsp;Jeong Rye Park ,&nbsp;Jongyook Park","doi":"10.1016/j.ejc.2025.104121","DOIUrl":"10.1016/j.ejc.2025.104121","url":null,"abstract":"&lt;div&gt;&lt;div&gt;For a positive integer &lt;span&gt;&lt;math&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, a putative strongly regular graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; with parameters &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;32&lt;/mn&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; satisfies both the Krein condition and the absolute bound. Also the multiplicities of the eigenvalues of the graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; are integers. This may mean that such a strongly regular graph exists. However, Koolen and Gebremichel proved that such a strongly regular graph does not exist for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, we generalize their method for all &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and rule out the infinite family of such strongly regular graphs. In order to do so, we find a restriction on the orders of two large maximal cliques intersecting in many vertices. And we also look at the case where the equality of the claw-bound holds to find an upper bound on the order of a coclique in a local graph (when &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is not Terwilliger). In a similar fashion, we note that one can also rule out another infinite family of putative strongly regular graphs with parameters &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;32&lt;/mn&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;64&lt;/mn&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;44&lt;/mn&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mr","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104121"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Counting sparse induced subgraphs in locally dense graphs","authors":"Rajko Nenadov","doi":"10.1016/j.ejc.2025.104125","DOIUrl":"10.1016/j.ejc.2025.104125","url":null,"abstract":"<div><div>An <span><math><mi>n</mi></math></span>-vertex graph <span><math><mi>G</mi></math></span> is locally dense if every induced subgraph of size larger than <span><math><mrow><mi>ζ</mi><mi>n</mi></mrow></math></span> has density at least <span><math><mrow><mi>d</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>, for some parameters <span><math><mrow><mi>ζ</mi><mo>,</mo><mi>d</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>. We show that the number of induced subgraphs of <span><math><mi>G</mi></math></span> with <span><math><mi>m</mi></math></span> vertices and maximum degree significantly smaller than <span><math><mrow><mi>d</mi><mi>m</mi></mrow></math></span> is roughly <span><math><mfenced><mrow><mfrac><mrow><mi>ζ</mi><mi>n</mi></mrow><mrow><mi>m</mi></mrow></mfrac></mrow></mfenced></math></span>, for <span><math><mrow><mi>m</mi><mo>≪</mo><mi>ζ</mi><mi>n</mi></mrow></math></span> which is not too small. This generalises a result of Kohayakawa, Lee, Rödl, and Samotij on the number of independent sets in locally dense graphs. As an application, we slightly improve a result of Balogh, Chen, and Luo on the generalised Erdős–Rogers function for graphs with small extremal number.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104125"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A combinatorial characterization of S2 binomial edge ideals","authors":"Davide Bolognini ,&nbsp;Antonio Macchia ,&nbsp;Giancarlo Rinaldo ,&nbsp;Francesco Strazzanti","doi":"10.1016/j.ejc.2025.104123","DOIUrl":"10.1016/j.ejc.2025.104123","url":null,"abstract":"<div><div>Several algebraic properties of a binomial edge ideal <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> can be interpreted in terms of combinatorial properties of its associated graph <span><math><mi>G</mi></math></span>. In particular, the so-called <em>cut sets</em> of a graph <span><math><mi>G</mi></math></span>, special sets of vertices that disconnect <span><math><mi>G</mi></math></span>, play an important role since they are in bijection with the minimal prime ideals of <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span>. In this paper we establish the first graph-theoretical characterization of binomial edge ideals <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> satisfying Serre’s condition <span><math><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></math></span> by proving that this is equivalent to having <span><math><mi>G</mi></math></span> <em>accessible</em>, which means that <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> is unmixed and the cut-point sets of <span><math><mi>G</mi></math></span> form an accessible set system. The proof relies on the combinatorial structure of the Stanley–Reisner simplicial complex of a multigraded generic initial ideal of <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span>, whose facets can be described in terms of cut-point sets. Another key step in the proof consists in proving the equivalence between accessibility and strong accessibility for the collection of cut sets of <span><math><mi>G</mi></math></span> with <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> unmixed. This result, interesting on its own, provides the first relevant class of set systems for which the previous two notions are equivalent.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104123"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143175394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}