European Journal of Combinatorics最新文献

{"title":"Coloring hypergraphs with excluded minors","authors":"Raphael Steiner","doi":"10.1016/j.ejc.2024.103971","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103971","url":null,"abstract":"<div><p>Hadwiger’s conjecture, among the most famous open problems in graph theory, states that every graph that does not contain <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> as a minor is properly <span><math><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-colorable.</p><p>The purpose of this work is to demonstrate that a natural extension of Hadwiger’s problem to hypergraph coloring exists, and to derive some first partial results and applications.</p><p>Generalizing ordinary graph minors to hypergraphs, we say that a hypergraph <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is a minor of a hypergraph <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, if a hypergraph isomorphic to <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> can be obtained from <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> via a finite sequence of the following operations:</p><p>• deleting vertices and hyperedges,</p><p>• contracting a hyperedge (i.e., merging the vertices of the hyperedge into a single vertex).</p><p>First we show that a weak extension of Hadwiger’s conjecture to hypergraphs holds true: For every <span><math><mrow><mi>t</mi><mo>≥</mo><mn>1</mn></mrow></math></span>, there exists a finite (smallest) integer <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> such that every hypergraph with no <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>-minor is <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span>-colorable, and we prove <span><math><mrow><mfenced><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></mfenced><mo>≤</mo><mi>h</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><mi>g</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> where <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> denotes the maximum chromatic number of graphs with no <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>-minor. Using the recent result by Delcourt and Postle that <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><mi>t</mi><mo>log</mo><mo>log</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span>, this yields <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><mi>t</mi><mo>log</mo><mo>log</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span>.</p><p>We further conjecture that <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mfenced><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103971"},"PeriodicalIF":1.0,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000568/pdfft?md5=7ddba04d4bd02c12e555b22107b8bb39&pid=1-s2.0-S0195669824000568-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140606814","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":"Special values of spectral zeta functions and combinatorics: Sturm–Liouville problems","authors":"Bing Xie ,&nbsp;Yigeng Zhao ,&nbsp;Yongqiang Zhao","doi":"10.1016/j.ejc.2024.103972","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103972","url":null,"abstract":"<div><p>In this paper, we apply the combinatorial results on counting permutations with fixed pinnacle and vale sets to evaluate the special values of the spectral zeta functions of Sturm–Liouville differential operators. As applications, we get a combinatorial formula for the special values of spectral zeta functions and give a new explicit formula for Bernoulli numbers.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103972"},"PeriodicalIF":1.0,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140558403","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":"A Brualdi–Hoffman–Turán problem on cycles","authors":"Xin Li ,&nbsp;Mingqing Zhai ,&nbsp;Jinlong Shu","doi":"10.1016/j.ejc.2024.103966","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103966","url":null,"abstract":"<div><p>Brualdi–Hoffman–Turán-type problem asks what is the maximum spectral radius <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of an <span><math><mi>H</mi></math></span>-free graph <span><math><mi>G</mi></math></span> on <span><math><mi>m</mi></math></span> edges? This problem gives a spectral perspective on the existence of a subgraph <span><math><mi>H</mi></math></span>. A significant result, due to Nikiforov, states that <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><msqrt><mrow><mn>2</mn><mi>m</mi><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>r</mi></mrow></mfrac><mo>)</mo></mrow></mrow></msqrt></mrow></math></span> for every <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-free graph <span><math><mi>G</mi></math></span> (Nikiforov, 2002). Bollobás and Nikiforov further conjectured <span><math><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><mi>m</mi><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>r</mi></mrow></mfrac><mo>)</mo></mrow></mrow></math></span> for every <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-free graph <span><math><mi>G</mi></math></span> (Bollobás and Nikiforov, 2007). Let <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> denote the graph obtained from a <span><math><mi>k</mi></math></span>-cycle by adding a chord between two vertices of distance two. Zhai, Lin and Shu conjectured that for <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and <span><math><mi>m</mi></math></span> sufficiently large, if <span><math><mi>G</mi></math></span> is a <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-free or <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span>-free graph, then <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>m</mi><mo>−</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>, with equality if and only if <span><math><mrow><mi>G</mi><mo>≅</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∇</mo><mrow><mo>(</mo><mfrac><mrow><mi>m</mi></","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103966"},"PeriodicalIF":1.0,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140550821","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":"Near classification of compact hyperbolic Coxeter d-polytopes with d+4 facets and related dimension bounds","authors":"Amanda Burcroff","doi":"10.1016/j.ejc.2024.103957","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103957","url":null,"abstract":"<div><p>We complete the classification of compact hyperbolic Coxeter <span><math><mi>d</mi></math></span>-polytopes with <span><math><mrow><mi>d</mi><mo>+</mo><mn>4</mn></mrow></math></span> facets for <span><math><mrow><mi>d</mi><mo>=</mo><mn>4</mn></mrow></math></span> and 5. By previous work of Felikson and Tumarkin, the only remaining dimension where new polytopes may arise is <span><math><mrow><mi>d</mi><mo>=</mo><mn>6</mn></mrow></math></span>. We derive a new method for generating the combinatorial types of these polytopes via the classification of point set order types. In dimensions 4 and 5, there are 348 and 51 polytopes, respectively, yielding many new examples for further study (also discovered independently by Ma and Zheng).</p><p>We furthermore provide new upper bounds on the dimension <span><math><mi>d</mi></math></span> of compact hyperbolic Coxeter polytopes with <span><math><mrow><mi>d</mi><mo>+</mo><mi>k</mi></mrow></math></span> facets for <span><math><mrow><mi>k</mi><mo>≤</mo><mn>10</mn></mrow></math></span>. It was shown by Vinberg in 1985 that for any <span><math><mi>k</mi></math></span>, we have <span><math><mrow><mi>d</mi><mo>≤</mo><mn>29</mn></mrow></math></span>, and no better bounds have previously been published for <span><math><mrow><mi>k</mi><mo>≥</mo><mn>5</mn></mrow></math></span>. As a consequence of our bounds, we prove that a compact hyperbolic Coxeter 29-polytope has at least 40 facets.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103957"},"PeriodicalIF":1.0,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140551697","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":"Poset modules of the 0-Hecke algebras and related quasisymmetric power sum expansions","authors":"Seung-Il Choi ,&nbsp;Young-Hun Kim ,&nbsp;Young-Tak Oh","doi":"10.1016/j.ejc.2024.103965","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103965","url":null,"abstract":"<div><p>Duchamp–Hivert–Thibon introduced the construction of a right <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></math></span>-module, denoted as <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span>, for any partial order <span><math><mi>P</mi></math></span> on the set <span><math><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></math></span>. This module is defined by specifying a suitable action of <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></math></span> on the set of linear extensions of <span><math><mi>P</mi></math></span>. In this paper, we refer to this module as the poset module associated with <span><math><mi>P</mi></math></span>. Firstly, we show that <span><math><mrow><msub><mrow><mo>⨁</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub><msub><mrow><mi>G</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> has a Hopf algebra structure that is isomorphic to the Hopf algebra of quasisymmetric functions, where <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> is the full subcategory of <span><math><mrow><mi>mod −</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></math></span> whose objects are direct sums of finitely many isomorphic copies of poset modules and <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> is the Grothendieck group of <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>. We also demonstrate how (anti-) automorphism twists interact with these modules, the induction product and restrictions. Secondly, we investigate the (type 1) quasisymmetric power sum expansion of some quasi-analogues <span><math><msub><mrow><mi>Y</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> of Schur functions, where <span><math><mi>α</mi></math></span> is a composition. We show that they can be expressed as the sum of the <span><math><mi>P</mi></math></span>-partition generating functions of specific posets, which allows us to utilize the result established by Liu–Weselcouch. Additionally, we provide a new algorithm for obtaining these posets. Using these findings, for the dual immaculate function and the extended Schur function, we express the coefficients appearing in the quasisymmetric power sum expansions in terms of border strip tableaux.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103965"},"PeriodicalIF":1.0,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140551698","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":"Cubic factor-invariant graphs of cycle quotient type—The alternating case","authors":"Brian Alspach ,&nbsp;Primož Šparl","doi":"10.1016/j.ejc.2024.103964","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103964","url":null,"abstract":"<div><p>We investigate connected cubic vertex-transitive graphs whose edge sets admit a partition into a 2-factor <span><math><mi>C</mi></math></span> and a 1-factor that is invariant under a vertex-transitive subgroup of the automorphism group of the graph and where the quotient graph with respect to <span><math><mi>C</mi></math></span> is a cycle. There are two essentially different types of such cubic graphs. In this paper we focus on the examples of what we call the alternating type. We classify all such examples admitting a vertex-transitive subgroup of the automorphism group of the graph preserving the corresponding 2-factor and also determine the ones for which the 2-factor is invariant under the full automorphism group of the graph. In this way we introduce a new infinite family of cubic vertex-transitive graphs that is a natural generalization of the well-known generalized Petersen graphs as well as of the honeycomb toroidal graphs. The family contains an infinite subfamily of arc-regular examples and an infinite subfamily of 2-arc-regular examples.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103964"},"PeriodicalIF":1.0,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000490/pdfft?md5=9aafdb85196268b7bd37d9ff8366aa0b&pid=1-s2.0-S0195669824000490-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140551696","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 lower bound on the multicolor size-Ramsey numbers of paths in hypergraphs","authors":"Deepak Bal ,&nbsp;Louis DeBiasio ,&nbsp;Allan Lo","doi":"10.1016/j.ejc.2024.103969","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103969","url":null,"abstract":"<div><p>The <span><math><mi>r</mi></math></span>-color size-Ramsey number of a <span><math><mi>k</mi></math></span>-uniform hypergraph <span><math><mi>H</mi></math></span>, denoted by <span><math><mrow><msub><mrow><mover><mrow><mi>R</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>, is the minimum number of edges in a <span><math><mi>k</mi></math></span>-uniform hypergraph <span><math><mi>G</mi></math></span> such that for every <span><math><mi>r</mi></math></span>-coloring of the edges of <span><math><mi>G</mi></math></span> there exists a monochromatic copy of <span><math><mi>H</mi></math></span>. In the case of 2-uniform paths <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, it is known that <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>n</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mover><mrow><mi>R</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><mrow><mo>(</mo><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>log</mo><mi>r</mi><mo>)</mo></mrow><mi>n</mi><mo>)</mo></mrow></mrow></math></span> with the best bounds essentially due to Krivelevich (2019). In a recent breakthrough result, Letzter et al. (2021) gave a linear upper bound on the <span><math><mi>r</mi></math></span>-color size-Ramsey number of the <span><math><mi>k</mi></math></span>-uniform tight path <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow><mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></msubsup></math></span>; i.e. <span><math><mrow><msub><mrow><mover><mrow><mi>R</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><msubsup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow><mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></msubsup><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>. At about the same time, Winter (2023) gave the first non-trivial lower bounds on the 2-color size-Ramsey number of <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow><mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></msubsup></math></span> for <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>; i.e. <span><math><mrow><msub><mrow><mover><mrow><mi>R</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><msubsup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow><mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow></msubsup><mo>)</mo></mrow><mo>≥</mo><mfrac><mrow><mn>8</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mi>n</mi><mo>−</mo><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103969"},"PeriodicalIF":1.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000544/pdfft?md5=8ed971a8a16a37bfa808eabee4a3a84c&pid=1-s2.0-S0195669824000544-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140549964","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 co-preLie structure from chronological loop erasure in graph walks","authors":"Loïc Foissy ,&nbsp;Pierre-Louis Giscard ,&nbsp;Cécile Mammez","doi":"10.1016/j.ejc.2024.103967","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103967","url":null,"abstract":"<div><p>We show that the chronological removal of cycles from a walk on a graph, known as Lawler’s loop-erasing procedure, generates a preLie co-algebra on the vector space spanned by the walks. In addition, we prove that the tensor and symmetric algebras of graph walks are Hopf algebras, provide their antipodes explicitly and recover the preLie co-algebra from a brace coalgebra on the tensor algebra of graph walks. Finally we exhibit sub-Hopf algebras associated to particular types of walks.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103967"},"PeriodicalIF":1.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000520/pdfft?md5=2816f59bd5b837c134bedca21dd85a3f&pid=1-s2.0-S0195669824000520-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140549963","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 sum-intersecting families of positive integers","authors":"Aaron Berger,&nbsp;Nitya Mani","doi":"10.1016/j.ejc.2024.103963","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103963","url":null,"abstract":"<div><p>We study the following natural arithmetic question regarding intersecting families: how large can a family of subsets of integers from <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mi>n</mi><mo>}</mo></mrow></math></span> be such that, for every pair of subsets in the family, the intersection contains a <em>sum</em> <span><math><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mi>z</mi></mrow></math></span>? We conjecture that any such <em>sum-intersecting</em> family must have size at most <span><math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mi>⋅</mi><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> (which would be tight if correct). Towards this conjecture, we show that every sum-intersecting family has at most <span><math><mrow><mn>0</mn><mo>.</mo><mn>32</mn><mi>⋅</mi><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> subsets.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103963"},"PeriodicalIF":1.0,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140543870","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":"A note on non-empty cross-intersecting families","authors":"Menglong Zhang,&nbsp;Tao Feng","doi":"10.1016/j.ejc.2024.103968","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103968","url":null,"abstract":"<div><p>The families <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊆</mo><mfenced><mrow><mfrac><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊆</mo><mfenced><mrow><mfrac><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>⊆</mo><mfenced><mrow><mfrac><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></math></span> are said to be cross-intersecting if <span><math><mrow><mrow><mo>|</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>|</mo></mrow><mo>⩾</mo><mn>1</mn></mrow></math></span> for any <span><math><mrow><mn>1</mn><mo>⩽</mo><mi>i</mi><mo>&lt;</mo><mi>j</mi><mo>⩽</mo><mi>r</mi></mrow></math></span> and <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></math></span>, <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></math></span>. Cross-intersecting families <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></math></span> are said to be <em>non-empty</em> if <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≠</mo><mo>0̸</mo></mrow></math></span> for any <span><math><mrow><mn>1</mn><mo>⩽</mo><mi>i</mi><mo>⩽</mo><mi>r</mi></mrow></math></span>. This paper shows that if <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊆</mo><mfenced><mrow><mfrac><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊆</mo><mfenced><mrow><mfrac><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>⊆</mo><mfenced><mrow><mfrac><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></math></span> are non-empty cross-intersecting families with <span><math><mrow><ms","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 103968"},"PeriodicalIF":1.0,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140535953","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}