{"title":"Pivot-minors and the Erdős-Hajnal conjecture","authors":"James Davies","doi":"10.1016/j.jctb.2025.04.004","DOIUrl":"10.1016/j.jctb.2025.04.004","url":null,"abstract":"<div><div>We prove a conjecture of Kim and Oum that every proper pivot-minor-closed class of graphs has the strong Erdős-Hajnal property. More precisely, for every graph <em>H</em>, there exists <span><math><mi>ϵ</mi><mo>></mo><mn>0</mn></math></span> such that every <em>n</em>-vertex graph with no pivot-minor isomorphic to <em>H</em> contains two sets <span><math><mi>A</mi><mo>,</mo><mi>B</mi></math></span> of vertices such that <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>,</mo><mo>|</mo><mi>B</mi><mo>|</mo><mo>⩾</mo><mi>ϵ</mi><mi>n</mi></math></span> and <em>A</em> is complete or anticomplete to <em>B</em>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 257-278"},"PeriodicalIF":1.2,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143834331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rutger Campbell , J. Pascal Gollin , Kevin Hendrey , Raphael Steiner
{"title":"Optimal bounds for zero-sum cycles. I. Odd order","authors":"Rutger Campbell , J. Pascal Gollin , Kevin Hendrey , Raphael Steiner","doi":"10.1016/j.jctb.2025.04.003","DOIUrl":"10.1016/j.jctb.2025.04.003","url":null,"abstract":"<div><div>For a finite (not necessarily abelian) group <span><math><mo>(</mo><mi>Γ</mi><mo>,</mo><mo>⋅</mo><mo>)</mo></math></span>, let <span><math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> denote the smallest positive integer <em>n</em> such that for each labelling of the arcs of the complete digraph of order <em>n</em> using elements from Γ, there exists a directed cycle such that the arc-labels along the cycle multiply to the identity. Alon and Krivelevich <span><span>[2]</span></span> initiated the study of the parameter <span><math><mi>n</mi><mo>(</mo><mo>⋅</mo><mo>)</mo></math></span> on cyclic groups and proved <span><math><mi>n</mi><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><mi>q</mi><mi>log</mi><mo></mo><mi>q</mi><mo>)</mo></math></span>. This was later improved to a linear bound of <span><math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>≤</mo><mn>8</mn><mo>|</mo><mi>Γ</mi><mo>|</mo></math></span> for every finite abelian group by Mészáros and the last author <span><span>[8]</span></span>, and then further to <span><math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>≤</mo><mn>2</mn><mo>|</mo><mi>Γ</mi><mo>|</mo><mo>−</mo><mn>1</mn></math></span> for every non-trivial finite group independently by Berendsohn, Boyadzhiyska and Kozma <span><span>[3]</span></span> as well as by Akrami, Alon, Chaudhury, Garg, Mehlhorn and Mehta <span><span>[1]</span></span>.</div><div>In this series of two papers we conclude this line of research by proving that <span><math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>≤</mo><mo>|</mo><mi>Γ</mi><mo>|</mo><mo>+</mo><mn>1</mn></math></span> for every finite group <span><math><mo>(</mo><mi>Γ</mi><mo>,</mo><mo>⋅</mo><mo>)</mo></math></span>, which is the best possible such bound in terms of the group order and precisely determines the value of <span><math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> for all cyclic groups as <span><math><mi>n</mi><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>q</mi><mo>+</mo><mn>1</mn></math></span>.</div><div>In the present paper we prove the above result for all groups of odd order. The proof for groups of even order needs to overcome substantial additional obstacles and will be presented in the second part of this series.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 246-256"},"PeriodicalIF":1.2,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143834330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Embedding connected factorizations II","authors":"Amin Bahmanian , Anna Johnsen-Yu","doi":"10.1016/j.jctb.2025.03.003","DOIUrl":"10.1016/j.jctb.2025.03.003","url":null,"abstract":"<div><div>Let <span><math><mi>λ</mi><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span> be the complete <em>h</em>-uniform <em>n</em>-vertex hypergraph in which each edge is repeated <em>λ</em> times. For <span><math><mi>r</mi><mo>:</mo><mo>=</mo><mo>(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>, a <em>(partial)</em> <strong>r</strong><em>-factorization</em> of <span><math><mi>λ</mi><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span> is a partition of the edges of <span><math><mi>λ</mi><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span> into factors <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> such that each factor is spanning and the degree of all vertices in each <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is (at most) <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. Suppose that <span><math><mi>n</mi><mo>≥</mo><mo>(</mo><mi>h</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>. We establish necessary and sufficient conditions that ensure a partial <strong>r</strong>-factorization of <span><math><mi>λ</mi><msubsup><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span> can be embedded in a connected <strong>r</strong>-factorization of <span><math><mi>λ</mi><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span>. Moreover, we prove a general result which leads to a complete characterization of partial <span><math><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span>-factorizations of <em>any</em> sub-hypergraph of <span><math><mi>λ</mi><msubsup><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span> in connected <strong>r</strong>-factorizations of <span><math><mi>λ</mi><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span> so long as <em>q</em> meets a natural upper bound. These results can be seen as unified generalizations of many classical combinatorial results, and can also be restated as results on embedding partial symmetric latin hypercubes.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 374-398"},"PeriodicalIF":1.2,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143894863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A splitter theorem on 3-connected binary matroids and inner fans","authors":"João Paulo Costalonga","doi":"10.1016/j.jctb.2025.03.004","DOIUrl":"10.1016/j.jctb.2025.03.004","url":null,"abstract":"<div><div>We establish a splitter type theorem for 3-connected binary matroids regarding elements whose contraction preserves a fixed 3-connected minor and the vertical 3-connectivity. We established that, for 3-connected simple binary matroids <span><math><mi>N</mi><mo><</mo><mi>M</mi></math></span>, there is a disjoint family <span><math><mo>{</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo><mo>⊆</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>E</mi><mo>(</mo><mi>M</mi><mo>)</mo></mrow></msup></math></span> such that <span><math><mi>r</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>+</mo><mo>⋯</mo><mo>+</mo><mi>r</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>r</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><mo>⋯</mo><mo>∪</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>≥</mo><mi>r</mi><mo>(</mo><mi>M</mi><mo>)</mo><mo>−</mo><mi>r</mi><mo>(</mo><mi>N</mi><mo>)</mo></math></span>, each <span><math><mrow><mi>si</mi></mrow><mo>(</mo><mi>M</mi><mo>/</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> is 3-connected with an <em>N</em>-minor, and either <span><math><mo>|</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>=</mo><mn>1</mn></math></span> or <em>X</em> is a special type of fan. We also establish a stronger version of this result under specific hypotheses. These results have several consequences, including the generalizations for binary matroids of some results about contractible edges in 3-connected graphs and some other structural results for graphs and binary matroids.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 204-245"},"PeriodicalIF":1.2,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Micha Christoph, Charlotte Knierim, Anders Martinsson, Raphael Steiner
{"title":"Improved bounds for zero-sum cycles in Zpd","authors":"Micha Christoph, Charlotte Knierim, Anders Martinsson, Raphael Steiner","doi":"10.1016/j.jctb.2025.03.001","DOIUrl":"10.1016/j.jctb.2025.03.001","url":null,"abstract":"<div><div>For a finite abelian group <span><math><mo>(</mo><mi>Γ</mi><mo>,</mo><mo>+</mo><mo>)</mo></math></span>, let <span><math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> denote the smallest positive integer <em>n</em> such that for each labeling of the arcs of the complete digraph of order <em>n</em> using elements from Γ, there exists a directed cycle such that the total sum of the arc-labels along the cycle equals 0. Alon and Krivelevich initiated the study of the parameter <span><math><mi>n</mi><mo>(</mo><mo>⋅</mo><mo>)</mo></math></span> on cyclic groups and proved that <span><math><mi>n</mi><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><mi>q</mi><mi>log</mi><mo></mo><mi>q</mi><mo>)</mo></math></span>. Several improvements and generalizations of this bound have since been obtained, and an optimal bound in terms of the group order of the form <span><math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>≤</mo><mo>|</mo><mi>Γ</mi><mo>|</mo><mo>+</mo><mn>1</mn></math></span> was recently announced by Campbell, Gollin, Hendrey and the last author. While this bound is tight when the group Γ is cyclic, in cases when Γ is far from being cyclic, significant improvements on the bound can be made. In this direction, studying the prototypical case when <span><math><mi>Γ</mi><mo>=</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> is a power of a cyclic group of prime order, Letzter and Morrison [<em>Journal of Combinatorial Theory Series B, 2024</em>] showed that <span><math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mi>O</mi><mo>(</mo><mi>p</mi><mi>d</mi><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>d</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> and that <span><math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mi>O</mi><mo>(</mo><mi>d</mi><mi>log</mi><mo></mo><mi>d</mi><mo>)</mo></math></span>. They then posed the problem of proving an (asymptotically optimal) upper bound of <span><math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mi>O</mi><mo>(</mo><mi>p</mi><mi>d</mi><mo>)</mo></math></span> for all primes <em>p</em> and <span><math><mi>d</mi><mo>∈</mo><mi>N</mi></math></span>. In this paper, we solve this problem for <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> and improve their bound for all primes <span><math><mi>p</mi><mo>≥</mo><mn>3</mn></math></span> by proving <span><math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mn>5</mn><mi>d</mi></math></span> and <span><math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 365-373"},"PeriodicalIF":1.2,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143894864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sparse induced subgraphs of large treewidth","authors":"Édouard Bonnet","doi":"10.1016/j.jctb.2025.03.002","DOIUrl":"10.1016/j.jctb.2025.03.002","url":null,"abstract":"<div><div>Motivated by an induced counterpart of treewidth sparsifiers (i.e., sparse subgraphs keeping the treewidth large) provided by the celebrated Grid Minor theorem of Robertson and Seymour (1986) <span><span>[22]</span></span> or by a classic result of Chekuri and Chuzhoy (2015) <span><span>[5]</span></span>, we show that for any natural numbers <em>t</em> and <em>w</em>, and real <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span>, there is an integer <span><math><mi>W</mi><mo>:</mo><mo>=</mo><mi>W</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>w</mi><mo>,</mo><mi>ε</mi><mo>)</mo></math></span> such that every graph with treewidth at least <em>W</em> and no <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span> subgraph admits a 2-connected <em>n</em>-vertex induced subgraph with treewidth at least <em>w</em> and at most <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mo>)</mo><mi>n</mi></math></span> edges. The induced subgraph is either a subdivided wall, or its line graph, or a spanning supergraph of a subdivided biclique. This in particular extends a result of Weißauer (2019) <span><span>[25]</span></span> that graphs of large treewidth have a large biclique subgraph or a long induced cycle.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 184-203"},"PeriodicalIF":1.2,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143685958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some results and problems on tournament structure","authors":"Tung Nguyen , Alex Scott , Paul Seymour","doi":"10.1016/j.jctb.2025.02.002","DOIUrl":"10.1016/j.jctb.2025.02.002","url":null,"abstract":"<div><div>This paper is a survey of results and problems related to the following question: is it true that if <em>G</em> is a tournament with sufficiently large chromatic number, then <em>G</em> has two vertex-disjoint subtournaments <span><math><mi>A</mi><mo>,</mo><mi>B</mi></math></span>, both with large chromatic number, such that all edges between them are directed from <em>A</em> to <em>B</em>? We describe what we know about this question, and report some progress on several other related questions, on tournament colouring and domination.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 146-183"},"PeriodicalIF":1.2,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Ramsey numbers of bounded degree trees versus general graphs","authors":"Richard Montgomery , Matías Pavez-Signé , Jun Yan","doi":"10.1016/j.jctb.2025.02.004","DOIUrl":"10.1016/j.jctb.2025.02.004","url":null,"abstract":"<div><div>For every <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and Δ, we prove that there exists a constant <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>Δ</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> such that the following holds. For every graph <em>H</em> with <span><math><mi>χ</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>=</mo><mi>k</mi></math></span> and every tree <em>T</em> with at least <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>Δ</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>|</mo><mi>H</mi><mo>|</mo></math></span> vertices and maximum degree at most Δ, the Ramsey number <span><math><mi>R</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is <span><math><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mo>|</mo><mi>T</mi><mo>|</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>σ</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>, where <span><math><mi>σ</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> is the size of a smallest colour class across all proper <em>k</em>-colourings of <em>H</em>. This is tight up to the value of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>Δ</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span>, and confirms a conjecture of Balla, Pokrovskiy, and Sudakov.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 102-145"},"PeriodicalIF":1.2,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143453850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Tree amalgamations and quasi-isometries","authors":"Matthias Hamann","doi":"10.1016/j.jctb.2025.02.003","DOIUrl":"10.1016/j.jctb.2025.02.003","url":null,"abstract":"<div><div>We investigate the connections between tree amalgamations and quasi-isometries. In particular, we prove that the quasi-isometry type of multi-ended accessible quasi-transitive connected locally finite graphs is determined by the quasi-isometry type of their one-ended factors in any of their terminal factorisations. Our results carry over theorems of Papasoglu and Whyte on quasi-isometries between multi-ended groups to those between multi-ended graphs. In the end, we discuss the impact of our results to a question of Woess.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 83-101"},"PeriodicalIF":1.2,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143419816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A counterexample to the coarse Menger conjecture","authors":"Tung Nguyen , Alex Scott , Paul Seymour","doi":"10.1016/j.jctb.2025.01.004","DOIUrl":"10.1016/j.jctb.2025.01.004","url":null,"abstract":"<div><div>Menger's well-known theorem from 1927 characterizes when it is possible to find <em>k</em> vertex-disjoint paths between two sets of vertices in a graph <em>G</em>. Recently, Georgakopoulos and Papasoglu and, independently, Albrechtsen, Huynh, Jacobs, Knappe and Wollan conjectured a coarse analogue of Menger's theorem, when the <em>k</em> paths are required to be pairwise at some distance at least <em>d</em>. The result is known for <span><math><mi>k</mi><mo>≤</mo><mn>2</mn></math></span>, but we will show that it is false for all <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, even if <em>G</em> is constrained to have maximum degree at most three. We also give a simpler proof of the result when <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 68-82"},"PeriodicalIF":1.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143403450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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