European Journal of Combinatorics最新文献

{"title":"Spanning hypertrees, vertex tours and meanders","authors":"Robert Cori ,&nbsp;Gábor Hetyei","doi":"10.1016/j.ejc.2023.103805","DOIUrl":"10.1016/j.ejc.2023.103805","url":null,"abstract":"<div><p>This paper revisits the notion of a spanning hypertree of a hypermap introduced by one of its authors and shows that it allows to shed new light on a very diverse set of recent results.</p><p><span><span>The tour of a map along one of its spanning trees used by Bernardi may be generalized to hypermaps and we show that it is equivalent to a dual tour described by Cori (1976) and Machì(1982). We give a bijection between the spanning hypertrees of the reciprocal of the </span>plane graph with 2 vertices and </span><span><math><mi>n</mi></math></span> parallel edges and the meanders of order <span><math><mi>n</mi></math></span> and a bijection of the same kind between semimeanders of order <span><math><mi>n</mi></math></span> and spanning hypertrees of the reciprocal of a plane graph with a single vertex and <span><math><mrow><mi>n</mi><mo>/</mo><mn>2</mn></mrow></math></span> nested edges. We introduce hyperdeletions and hypercontractions in a hypermap which allow to count the spanning hypertrees of a hypermap recursively, and create a link with the computation of the Tutte polynomial of a graph. Having a particular interest in hypermaps which are reciprocals of maps, we generalize the reduction map introduced by Franz and Earnshaw to enumerate meanders to a reduction map that allows the enumeration of the spanning hypertrees of such hypermaps.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103805"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119581","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":"Quasiplanar graphs, string graphs, and the Erdős–Gallai problem","authors":"Jacob Fox ,&nbsp;János Pach ,&nbsp;Andrew Suk","doi":"10.1016/j.ejc.2023.103811","DOIUrl":"10.1016/j.ejc.2023.103811","url":null,"abstract":"<div><p>An <span><math><mi>r</mi></math></span>-<em>quasiplanar graph</em> is a graph drawn in the plane with no <span><math><mi>r</mi></math></span> pairwise crossing edges. Let <span><math><mrow><mi>s</mi><mo>≥</mo><mn>3</mn></mrow></math></span> be an integer and <span><math><mrow><mi>r</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>s</mi></mrow></msup></mrow></math></span>. We prove that there is a constant <span><math><mi>C</mi></math></span> such that every <span><math><mi>r</mi></math></span>-quasiplanar graph with <span><math><mrow><mi>n</mi><mo>≥</mo><mi>r</mi></mrow></math></span> vertices has at most <span><math><mrow><mi>n</mi><msup><mrow><mfenced><mrow><mi>C</mi><msup><mrow><mi>s</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>log</mo><mi>n</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>s</mi><mo>−</mo><mn>4</mn></mrow></msup></mrow></math></span> edges.</p><p>A graph whose vertices are continuous curves in the plane, two being connected by an edge if and only if they intersect, is called a <em>string graph</em>. We show that for every <span><math><mrow><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>, there exists <span><math><mrow><mi>δ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> such that every string graph with <span><math><mi>n</mi></math></span> vertices whose chromatic number is at least <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>ϵ</mi></mrow></msup></math></span> contains a clique of size at least <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>δ</mi></mrow></msup></math></span>. A clique of this size or a coloring using fewer than <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>ϵ</mi></mrow></msup></math></span> colors can be found by a polynomial time algorithm in terms of the size of the geometric representation of the set of strings.</p><p>In the process, we use, generalize, and strengthen previous results of Lee, Tomon, and others. All of our theorems are related to geometric variants of the following classical graph-theoretic problem of Erdős, Gallai, and Rogers. Given a <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>-free graph on <span><math><mi>n</mi></math></span> vertices and an integer <span><math><mrow><mi>s</mi><mo>&lt;</mo><mi>r</mi></mrow></math></span>, at least how many vertices can we find such that the subgraph induced by them is <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>-free?</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103811"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669823001282/pdfft?md5=d736f71ea441144851fb043750102221&pid=1-s2.0-S0195669823001282-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135607241","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 few words about maps","authors":"Robert Cori ,&nbsp;Yiting Jiang ,&nbsp;Patrice Ossona de Mendez ,&nbsp;Pierre Rosenstiehl","doi":"10.1016/j.ejc.2023.103810","DOIUrl":"10.1016/j.ejc.2023.103810","url":null,"abstract":"<div><p><span><span>In this paper, we survey some properties, encoding, and bijections involving combinatorial maps, double occurrence words, and chord diagrams. We particularly study quasi-trees from a purely combinatorial point of view and derive a topological representation of maps with a given spanning quasi-tree using two fundamental polygons, which extends the representation of planar maps based on the equivalence with bipartite </span>circle graphs. Then, we focus on Depth-First Search trees and their connection with a </span>poset we define on the spanning quasi-trees of a map. We apply the bijections obtained in the first section to the problem of enumerating loopless rooted maps. Finally, we return to the planar case and discuss a decomposition of planar rooted loopless maps and its consequences on planar rooted loopless map enumeration.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103810"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135605805","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":"Some memories","authors":"Marc Bousset ,&nbsp;Michel Imbert ,&nbsp;Armelle Vanot ,&nbsp;Philippe Gallic","doi":"10.1016/j.ejc.2023.103818","DOIUrl":"10.1016/j.ejc.2023.103818","url":null,"abstract":"","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103818"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138536323","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":"Hereditary classes of ordered sets of width at most two","authors":"Maurice Pouzet ,&nbsp;Imed Zaguia","doi":"10.1016/j.ejc.2023.103813","DOIUrl":"10.1016/j.ejc.2023.103813","url":null,"abstract":"<div><p>This paper is a contribution to the study of hereditary classes of relational structures, these classes being quasi-ordered by embeddability. It deals with the specific case of ordered sets of width two and the corresponding bichains and incomparability graphs.</p><p>Several open problems about hereditary classes of relational structures which have been considered over the years have a positive answer in this case. For example, well-quasi-ordered hereditary classes of finite bipartite permutation graphs, respectively finite 321-avoiding permutations, have been characterized by Korpelainen, Lozin and Mayhill, respectively by Albert, Brignall, Ruškuc and Vatter.</p><p>In this paper we present an overview of properties of these hereditary classes in the framework of the Theory of Relations as presented by Roland Fraïssé.</p><p>We provide another proof of the results mentioned above. It is based on the existence of a countable universal poset of width two, obtained by the first author in 1978, his notion of multichainability (1978) (a kind of analog to letter-graphs), and metric properties of incomparability graphs. Using Laver’s theorem (1971) on better-quasi-ordering (bqo) of countable chains we prove that a wqo hereditary class of finite or countable bipartite permutation graphs is necessarily bqo. This gives a positive answer to a conjecture of Nash-Williams (1965) in this case. We extend a previous result of Albert et al. by proving that if a hereditary class of finite, respectively countable, bipartite permutation graphs is wqo, respectively bqo, then the corresponding hereditary classes of posets of width at most two and bichains are wqo, respectively bqo.</p><p>Several notions of labelled wqo are also considered. We prove that they are all equivalent in the case of bipartite permutation graphs, posets of width at most two and the corresponding bichains. We characterize hereditary classes of finite bipartite permutation graphs which remain wqo when labels from a wqo are added. Hereditary classes of posets of width two, bipartite permutation graphs and the corresponding bichains having finitely many bounds are also characterized.</p><p><span>We prove that a hereditary class of finite bipartite permutation graphs is not wqo if and only if it embeds the poset of finite subsets of </span><span><math><mi>N</mi></math></span> ordered by set inclusion. This answers a long standing conjecture of the first author in the case of bipartite permutation graphs.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103813"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134934369","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":"Pendant appearances and components in random graphs from structured classes","authors":"Colin McDiarmid","doi":"10.1016/j.ejc.2024.103994","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103994","url":null,"abstract":"<div><p>We consider random graphs sampled uniformly from a structured class of graphs, such as the class of graphs embeddable in a given surface. We sharpen earlier results on pendant appearances, concerning for example numbers of leaves, and we find the asymptotic distribution of components other than the giant component, under quite general conditions.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103994"},"PeriodicalIF":1.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000799/pdfft?md5=579bb065983d305f54b0cc541656b245&pid=1-s2.0-S0195669824000799-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141242537","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":"(−1)-enumerations of arrowed Gelfand–Tsetlin patterns","authors":"Ilse Fischer,&nbsp;Florian Schreier-Aigner","doi":"10.1016/j.ejc.2024.103979","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103979","url":null,"abstract":"<div><p>Arrowed Gelfand–Tsetlin patterns have recently been introduced to study alternating sign matrices. In this paper, we show that a <span><math><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-enumeration of arrowed Gelfand–Tsetlin patterns can be expressed by a simple product formula. The numbers are up to <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span> a one-parameter generalization of the numbers <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><mn>2</mn></mrow></msup><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mfrac><mrow><mrow><mo>(</mo><mn>4</mn><mi>j</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mo>!</mo></mrow><mrow><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mi>j</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>!</mo></mrow></mfrac></mrow></math></span> that appear in recent work of Di Francesco. A second result concerns the <span><math><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-enumeration of arrowed Gelfand–Tsetlin patterns when excluding double-arrows as decoration in which case we also obtain a simple product formula. We are also able to provide signless interpretations of our results. The proofs of the enumeration formulas are based on a recent Littlewood-type identity, which allows us to reduce the problem to the evaluations of two determinants. The evaluations are accomplished by means of the LU-decompositions of the underlying matrices, and an extension of Sister Celine’s algorithm as well as creative telescoping to evaluate certain triple sums. In particular, we use implementations of such algorithms by Koutschan, and by Wegschaider and Riese.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103979"},"PeriodicalIF":1.0,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000647/pdfft?md5=55104cbe526326423121d99e38e209de&pid=1-s2.0-S0195669824000647-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141242536","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":"Mutual-visibility problems on graphs of diameter two","authors":"Serafino Cicerone ,&nbsp;Gabriele Di Stefano ,&nbsp;Sandi Klavžar ,&nbsp;Ismael G. Yero","doi":"10.1016/j.ejc.2024.103995","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103995","url":null,"abstract":"<div><p>The mutual-visibility problem in a graph <span><math><mi>G</mi></math></span> asks for the cardinality of a largest set of vertices <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> so that for any two vertices <span><math><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi></mrow></math></span> there is a shortest <span><math><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></math></span>-path <span><math><mi>P</mi></math></span> so that all internal vertices of <span><math><mi>P</mi></math></span> are not in <span><math><mi>S</mi></math></span>. This is also said as <span><math><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></math></span> are visible with respect to <span><math><mi>S</mi></math></span>, or <span><math><mi>S</mi></math></span>-visible for short. Variations of this problem are known, based on the extension of the visibility property of vertices that are in and/or outside <span><math><mi>S</mi></math></span>. Such variations are called total, outer and dual mutual-visibility problems. This work is focused on studying the corresponding four visibility parameters in graphs of diameter two, throughout showing bounds and/or closed formulae for these parameters.</p><p>The mutual-visibility problem in the Cartesian product of two complete graphs is equivalent to (an instance of) the celebrated Zarankiewicz’s problem. Here we study the dual and outer mutual-visibility problem for the Cartesian product of two complete graphs and all the mutual-visibility problems for the direct product of such graphs as well. We also study all the mutual-visibility problems for the line graphs of complete and complete bipartite graphs. As a consequence of this study, we present several relationships between the mentioned problems and some instances of the classical Turán problem. Moreover, we study the visibility problems for cographs and several non-trivial diameter-two graphs of minimum size.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103995"},"PeriodicalIF":1.0,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000805/pdfft?md5=8b2737f5ff200dbacddcfc4e58622e8e&pid=1-s2.0-S0195669824000805-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141090829","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":"Distinct distances in R3 between quadratic and orthogonal curves","authors":"Toby Aldape ,&nbsp;Jingyi Liu ,&nbsp;Gregory Pylypovych ,&nbsp;Adam Sheffer ,&nbsp;Minh-Quan Vo","doi":"10.1016/j.ejc.2024.103993","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103993","url":null,"abstract":"<div><p>We study the minimum number of distinct distances between point sets on two curves in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. Assume that one curve contains <span><math><mi>m</mi></math></span> points and the other <span><math><mi>n</mi></math></span> points. Our main results:</p><p>(a) When the curves are conic sections, we characterize all cases where the number of distances is <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>. This includes new constructions for points on two parabolas, two ellipses, and one ellipse and one hyperbola. In all other cases, the number of distances is <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><mo>min</mo><mrow><mo>{</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>,</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>.</p><p>(b) When the curves are not necessarily algebraic but smooth and contained in perpendicular planes, we characterize all cases where the number of distances is <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>. This includes a surprising new construction of non-algebraic curves that involve logarithms. In all other cases, the number of distances is <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><mo>min</mo><mrow><mo>{</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>,</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103993"},"PeriodicalIF":1.0,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141068815","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":"Variations on the Bollobás set-pair theorem","authors":"Gábor Hegedüs ,&nbsp;Péter Frankl","doi":"10.1016/j.ejc.2024.103983","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103983","url":null,"abstract":"<div><p>Let <span><math><mi>X</mi></math></span> be an <span><math><mi>n</mi></math></span>-element set. A set-pair system <span><math><mrow><mi>P</mi><mo>=</mo><msub><mrow><mrow><mo>{</mo><mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>}</mo></mrow></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>m</mi></mrow></msub></mrow></math></span> is a collection of pairs of disjoint subsets of <span><math><mi>X</mi></math></span>. It is called skew Bollobás system if <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≠</mo><mo>0̸</mo></mrow></math></span> for all <span><math><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>&lt;</mo><mi>j</mi><mo>≤</mo><mi>m</mi></mrow></math></span>. The best possible inequality <span><math><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><mfrac><mrow><mn>1</mn></mrow><mrow><mfenced><mfrac><mrow><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow><mo>+</mo><mrow><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mrow><mrow><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mrow></mfrac></mfenced></mrow></mfrac><mo>≤</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>.</mo></mrow></math></span> is established along with some more results of similar flavor.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103983"},"PeriodicalIF":1.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000684/pdfft?md5=21ca6f2c126894c4d8c03cff15a181b3&pid=1-s2.0-S0195669824000684-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140902015","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}