Pub Date : 2025-02-06DOI: 10.1016/j.ejc.2025.104131
Laura Grave de Peralta , Inga Valentiner-Branth
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 , -spherical and purely -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 , a case that was not covered in earlier works.
{"title":"High-dimensional expanders from Kac–Moody–Steinberg groups","authors":"Laura Grave de Peralta , 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}
Pub Date : 2025-02-05DOI: 10.1016/j.ejc.2025.104128
Yongchun Zang , Robin D.P. Zhou
Ascent sequences were introduced by Bousquet-Mélou, Claesson, Dukes and Kitaev, and are in bijection with unlabeled -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 -ascent sequences. They showed that some maps from the weak case can be extended to bijections for general 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 permutations and a class of factorial posets which we call difference posets, both of which are shown to be in bijection with -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.
{"title":"Difference ascent sequences and related combinatorial structures","authors":"Yongchun Zang , 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}
Pub Date : 2025-01-30DOI: 10.1016/j.ejc.2025.104124
L’ubomíra Dvořáková , Karel Klouda , Edita Pelantová
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 -ary sequences with , 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 -ary sequences, with , is equal to two, independently of the alphabet size.
{"title":"The asymptotic repetition threshold of sequences rich in palindromes","authors":"L’ubomíra Dvořáková , Karel Klouda , 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}
Pub Date : 2025-01-28DOI: 10.1016/j.ejc.2025.104121
Jack H. Koolen , Brhane Gebremichel , Jeong Rye Park , Jongyook Park
<div><div>For a positive integer <span><math><mi>t</mi></math></span>, a putative strongly regular graph <span><math><mi>G</mi></math></span> with parameters <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>k</mi><mo>+</mo><mfrac><mrow><mi>k</mi><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo>)</mo></mrow></mrow><mrow><mi>μ</mi></mrow></mfrac><mo>,</mo><mn>2</mn><mi>t</mi><mrow><mo>(</mo><mn>4</mn><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mi>μ</mi><mo>,</mo><mrow><mo>(</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>32</mn><msup><mrow><mi>t</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mn>4</mn><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>8</mn><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> satisfies both the Krein condition and the absolute bound. Also the multiplicities of the eigenvalues of the graph <span><math><mi>G</mi></math></span> 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 <span><math><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow></math></span>. In this paper, we generalize their method for all <span><math><mrow><mi>t</mi><mo>≥</mo><mn>1</mn></mrow></math></span> 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 <span><math><mi>G</mi></math></span> 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 <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>k</mi><mo>+</mo><mfrac><mrow><mi>k</mi><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo>)</mo></mrow></mrow><mrow><mi>μ</mi></mrow></mfrac><mo>,</mo><mrow><mo>(</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>4</mn><mi>t</mi><mo>+</mo><mn>3</mn><mo>)</mo></mrow><mi>μ</mi><mo>,</mo><mrow><mo>(</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>32</mn><msup><mrow><mi>t</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mn>64</mn><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>44</mn><mi>t</mi><mo>+</mo><mn>9</mn><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>)</mo></mr
{"title":"Non-existence of two infinite families of strongly regular graphs","authors":"Jack H. Koolen , Brhane Gebremichel , Jeong Rye Park , Jongyook Park","doi":"10.1016/j.ejc.2025.104121","DOIUrl":"10.1016/j.ejc.2025.104121","url":null,"abstract":"<div><div>For a positive integer <span><math><mi>t</mi></math></span>, a putative strongly regular graph <span><math><mi>G</mi></math></span> with parameters <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>k</mi><mo>+</mo><mfrac><mrow><mi>k</mi><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo>)</mo></mrow></mrow><mrow><mi>μ</mi></mrow></mfrac><mo>,</mo><mn>2</mn><mi>t</mi><mrow><mo>(</mo><mn>4</mn><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mi>μ</mi><mo>,</mo><mrow><mo>(</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>32</mn><msup><mrow><mi>t</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mn>4</mn><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>8</mn><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> satisfies both the Krein condition and the absolute bound. Also the multiplicities of the eigenvalues of the graph <span><math><mi>G</mi></math></span> 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 <span><math><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow></math></span>. In this paper, we generalize their method for all <span><math><mrow><mi>t</mi><mo>≥</mo><mn>1</mn></mrow></math></span> 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 <span><math><mi>G</mi></math></span> 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 <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>k</mi><mo>+</mo><mfrac><mrow><mi>k</mi><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo>)</mo></mrow></mrow><mrow><mi>μ</mi></mrow></mfrac><mo>,</mo><mrow><mo>(</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>4</mn><mi>t</mi><mo>+</mo><mn>3</mn><mo>)</mo></mrow><mi>μ</mi><mo>,</mo><mrow><mo>(</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>32</mn><msup><mrow><mi>t</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mn>64</mn><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>44</mn><mi>t</mi><mo>+</mo><mn>9</mn><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>)</mo></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}
Pub Date : 2025-01-28DOI: 10.1016/j.ejc.2025.104125
Rajko Nenadov
An -vertex graph is locally dense if every induced subgraph of size larger than has density at least , for some parameters . We show that the number of induced subgraphs of with vertices and maximum degree significantly smaller than is roughly , for 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.
{"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>></mo><mn>0</mn></mrow></math></span>, for some parameters <span><math><mrow><mi>ζ</mi><mo>,</mo><mi>d</mi><mo>></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}
Pub Date : 2025-01-28DOI: 10.1016/j.ejc.2025.104123
Davide Bolognini , Antonio Macchia , Giancarlo Rinaldo , Francesco Strazzanti
Several algebraic properties of a binomial edge ideal can be interpreted in terms of combinatorial properties of its associated graph . In particular, the so-called cut sets of a graph , special sets of vertices that disconnect , play an important role since they are in bijection with the minimal prime ideals of . In this paper we establish the first graph-theoretical characterization of binomial edge ideals satisfying Serre’s condition by proving that this is equivalent to having accessible, which means that is unmixed and the cut-point sets of 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 , 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 with unmixed. This result, interesting on its own, provides the first relevant class of set systems for which the previous two notions are equivalent.
{"title":"A combinatorial characterization of S2 binomial edge ideals","authors":"Davide Bolognini , Antonio Macchia , Giancarlo Rinaldo , 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}
Pub Date : 2025-01-28DOI: 10.1016/j.ejc.2025.104120
Sofiya Burova , Lyuben Lichev
The online semi-random graph process is a one-player game which starts with the empty graph on vertices. At every round, a player (called Builder) is presented with a vertex chosen uniformly at random and independently from previous rounds, and constructs an edge of their choice that is incident to . Inspired by recent advances on the semi-random graph process, we define a family of generalized online semi-random models.
We analyse a particular instance that shares similar features with the original semi-random graph process and determine the hitting times of the classical graph properties minimum degree , -connectivity, containment of a perfect matching, a Hamiltonian cycle and an -factor for a fixed graph possessing an additional tree-like property. Along the way, we derive a few consequences of the famous Aldous-Broder algorithm that may be of independent interest.
{"title":"The semi-random tree process","authors":"Sofiya Burova , Lyuben Lichev","doi":"10.1016/j.ejc.2025.104120","DOIUrl":"10.1016/j.ejc.2025.104120","url":null,"abstract":"<div><div>The online semi-random graph process is a one-player game which starts with the empty graph on <span><math><mi>n</mi></math></span> vertices. At every round, a player (called Builder) is presented with a vertex <span><math><mi>v</mi></math></span> chosen uniformly at random and independently from previous rounds, and constructs an edge of their choice that is incident to <span><math><mi>v</mi></math></span>. Inspired by recent advances on the semi-random graph process, we define a family of generalized online semi-random models.</div><div>We analyse a particular instance that shares similar features with the original semi-random graph process and determine the hitting times of the classical graph properties minimum degree <span><math><mi>k</mi></math></span>, <span><math><mi>k</mi></math></span>-connectivity, containment of a perfect matching, a Hamiltonian cycle and an <span><math><mi>H</mi></math></span>-factor for a fixed graph <span><math><mi>H</mi></math></span> possessing an additional tree-like property. Along the way, we derive a few consequences of the famous Aldous-Broder algorithm that may be of independent interest.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104120"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176371","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}
Pub Date : 2025-01-24DOI: 10.1016/j.ejc.2025.104122
Alan Frieze , Pu Gao , Calum MacRury , Paweł Prałat , Gregory B. Sorkin
The semi-random graph process is an adaptive random graph process in which an online algorithm is initially presented an empty graph on vertices. In each round, a vertex is presented to the algorithm independently and uniformly at random. The algorithm then adaptively selects a vertex , and adds the edge to the graph. For a given graph property, the objective of the algorithm is to force the graph to satisfy this property asymptotically almost surely in as few rounds as possible.
We focus on the property of Hamiltonicity. We present an adaptive strategy which creates a Hamiltonian cycle in rounds, where is derived from the solution to a system of differential equations. We also show that achieving Hamiltonicity requires at least rounds, where .
{"title":"Building Hamiltonian cycles in the semi-random graph process in less than 2n rounds","authors":"Alan Frieze , Pu Gao , Calum MacRury , Paweł Prałat , Gregory B. Sorkin","doi":"10.1016/j.ejc.2025.104122","DOIUrl":"10.1016/j.ejc.2025.104122","url":null,"abstract":"<div><div>The semi-random graph process is an adaptive random graph process in which an online algorithm is initially presented an empty graph on <span><math><mi>n</mi></math></span> vertices. In each round, a vertex <span><math><mi>u</mi></math></span> is presented to the algorithm independently and uniformly at random. The algorithm then adaptively selects a vertex <span><math><mi>v</mi></math></span>, and adds the edge <span><math><mrow><mi>u</mi><mi>v</mi></mrow></math></span> to the graph. For a given graph property, the objective of the algorithm is to force the graph to satisfy this property asymptotically almost surely in as few rounds as possible.</div><div>We focus on the property of Hamiltonicity. We present an adaptive strategy which creates a Hamiltonian cycle in <span><math><mrow><mi>α</mi><mi>n</mi></mrow></math></span> rounds, where <span><math><mrow><mi>α</mi><mo><</mo><mn>1</mn><mo>.</mo><mn>81696</mn></mrow></math></span> is derived from the solution to a system of differential equations. We also show that achieving Hamiltonicity requires at least <span><math><mrow><mi>β</mi><mi>n</mi></mrow></math></span> rounds, where <span><math><mrow><mi>β</mi><mo>></mo><mn>1</mn><mo>.</mo><mn>26575</mn></mrow></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104122"},"PeriodicalIF":1.0,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176369","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}
Pub Date : 2025-01-18DOI: 10.1016/j.ejc.2024.104118
Jack H. Koolen , Kefan Yu , Xiaoye Liang , Harrison Choi , Greg Markowsky
In this paper, we classify non-geometric distance-regular graphs of diameter at least 3 with smallest eigenvalue at least −3. This is progress towards what is hoped to be an eventual complete classification of distance-regular graphs with smallest eigenvalue at least −3, analogous to existing classification results available in the case that the smallest eigenvalue is at least −2.
{"title":"Non-geometric distance-regular graphs of diameter at least 3 with smallest eigenvalue at least −3","authors":"Jack H. Koolen , Kefan Yu , Xiaoye Liang , Harrison Choi , Greg Markowsky","doi":"10.1016/j.ejc.2024.104118","DOIUrl":"10.1016/j.ejc.2024.104118","url":null,"abstract":"<div><div>In this paper, we classify non-geometric distance-regular graphs of diameter at least 3 with smallest eigenvalue at least −3. This is progress towards what is hoped to be an eventual complete classification of distance-regular graphs with smallest eigenvalue at least −3, analogous to existing classification results available in the case that the smallest eigenvalue is at least −2.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104118"},"PeriodicalIF":1.0,"publicationDate":"2025-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176370","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}
Pub Date : 2025-01-16DOI: 10.1016/j.ejc.2024.104116
Meijie Lu , Xianchang Meng
In this paper, for any integer , we study the distribution of the visible lattice points in certain generalized Pólya walks on : perturbed Pólya walk and twisted Pólya walk. For the first case, we prove that the asymptotic proportion of visible points in a perturbed Pólya walk is almost surely , where denotes the Riemann zeta function. A trivial case of our result covers the standard Pólya walk. Moreover, we do numerical experiments for the second case, we conjecture that the proportion is also almost surely .
{"title":"Visible lattice points in Pólya’s walks","authors":"Meijie Lu , Xianchang Meng","doi":"10.1016/j.ejc.2024.104116","DOIUrl":"10.1016/j.ejc.2024.104116","url":null,"abstract":"<div><div>In this paper, for any integer <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, we study the distribution of the visible lattice points in certain generalized Pólya walks on <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>: perturbed Pólya walk and twisted Pólya walk. For the first case, we prove that the asymptotic proportion of visible points in a perturbed Pólya walk is almost surely <span><math><mrow><mn>1</mn><mo>/</mo><mi>ζ</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></math></span> denotes the Riemann zeta function. A trivial case of our result covers the standard Pólya walk. Moreover, we do numerical experiments for the second case, we conjecture that the proportion is also almost surely <span><math><mrow><mn>1</mn><mo>/</mo><mi>ζ</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104116"},"PeriodicalIF":1.0,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176366","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}
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