Journal of Combinatorial Theory Series B最新文献

{"title":"Ascending subgraph decomposition","authors":"Kyriakos Katsamaktsis ,&nbsp;Shoham Letzter ,&nbsp;Alexey Pokrovskiy ,&nbsp;Benny Sudakov","doi":"10.1016/j.jctb.2025.01.003","DOIUrl":"10.1016/j.jctb.2025.01.003","url":null,"abstract":"<div><div>A typical theme for many well-known decomposition problems is to show that some obvious necessary conditions for decomposing a graph <em>G</em> into copies of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> are also sufficient. One such problem was posed in 1987, by Alavi, Boals, Chartrand, Erdős, and Oellerman. They conjectured that the edges of every graph with <span><math><mo>(</mo><mtable><mtr><mtd><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></math></span> edges can be decomposed into subgraphs <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> such that each <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> has <em>i</em> edges and is isomorphic to a subgraph of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. In this paper we prove this conjecture for sufficiently large <em>m</em>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 14-44"},"PeriodicalIF":1.2,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386566","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 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>&lt;</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":"2025-07-01","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}

{"title":"Improved bounds for zero-sum cycles in Zpd","authors":"Micha Christoph,&nbsp;Charlotte Knierim,&nbsp;Anders Martinsson,&nbsp;Raphael Steiner","doi":"10.1016/j.jctb.2025.03.001","DOIUrl":"10.1016/j.jctb.2025.03.001","url":null,"abstract":"&lt;div&gt;&lt;div&gt;For a finite abelian group &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, let &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; denote the smallest positive integer &lt;em&gt;n&lt;/em&gt; such that for each labeling of the arcs of the complete digraph of order &lt;em&gt;n&lt;/em&gt; 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 &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; on cyclic groups and proved that &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Several improvements and generalizations of this bound have since been obtained, and an optimal bound in terms of the group order of the form &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; 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 &lt;span&gt;&lt;math&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; is a power of a cyclic group of prime order, Letzter and Morrison [&lt;em&gt;Journal of Combinatorial Theory Series B, 2024&lt;/em&gt;] showed that &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and that &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. They then posed the problem of proving an (asymptotically optimal) upper bound of &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for all primes &lt;em&gt;p&lt;/em&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, we solve this problem for &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and improve their bound for all primes &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; by proving &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 365-373"},"PeriodicalIF":1.2,"publicationDate":"2025-07-01","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>&gt;</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":"2025-07-01","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":"Dynamics of cycles in polyhedra I: The isolation lemma","authors":"Jan Kessler ,&nbsp;Jens M. Schmidt","doi":"10.1016/j.jctb.2024.03.008","DOIUrl":"10.1016/j.jctb.2024.03.008","url":null,"abstract":"&lt;div&gt;&lt;div&gt;A cycle &lt;em&gt;C&lt;/em&gt; of a graph &lt;em&gt;G&lt;/em&gt; is &lt;em&gt;isolating&lt;/em&gt; if every component of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; consists of a single vertex. We show that isolating cycles in polyhedral graphs can be extended to larger ones: every isolating cycle &lt;em&gt;C&lt;/em&gt; of length &lt;span&gt;&lt;math&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;⌊&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;⌋&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; implies an isolating cycle &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; of larger length that contains &lt;span&gt;&lt;math&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. By “hopping” iteratively to such larger cycles, we obtain a powerful and very general inductive motor for proving long cycles and computing them (we will give an algorithm with quadratic running time). This is the first step towards the so far elusive quest of finding a universal induction that captures longest cycles of polyhedral graph classes.&lt;/div&gt;&lt;div&gt;Our motor provides also a method to prove linear lower bounds on the length of Tutte cycles, as &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; will be a Tutte cycle of &lt;em&gt;G&lt;/em&gt; if &lt;em&gt;C&lt;/em&gt; is. We prove in addition that &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; if &lt;em&gt;G&lt;/em&gt;&lt;span&gt; contains no face of size five, which gives a new tool for results about cycle spectra, and provides evidence that faces of size five may obstruct many different cycle lengths. As a sample application, we test our motor on the following so far unsettled conjecture about essentially 4-connected graphs.&lt;/span&gt;&lt;/div&gt;&lt;div&gt;A planar graph is &lt;em&gt;essentially&lt;/em&gt; 4&lt;em&gt;-connected&lt;/em&gt; if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Essentially 4-connected graphs have been thoroughly investigated throughout literature as the subject of Hamiltonicity studies. Jackson and Wormald proved that every essentially 4-connected planar graph &lt;em&gt;G&lt;/em&gt; on &lt;em&gt;n&lt;/em&gt; vertices contains a cycle of length at least &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, and this result has recently been improved multiple times, culminating in the lower bound &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. However, the currently best known upper bound is given by an infinite family of such graphs in which no graph &lt;em&gt;G&lt;/em&gt; contains a cycle that is longer than &lt;","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 329-364"},"PeriodicalIF":1.2,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143894862","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 counterexample to the coarse Menger conjecture","authors":"Tung Nguyen ,&nbsp;Alex Scott ,&nbsp;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":"2025-07-01","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}

{"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":"2025-07-01","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":"Some results and problems on tournament structure","authors":"Tung Nguyen ,&nbsp;Alex Scott ,&nbsp;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":"2025-07-01","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":"Clustered coloring of (path + 2K1)-free graphs on surfaces","authors":"Zdeněk Dvořák","doi":"10.1016/j.jctb.2025.02.001","DOIUrl":"10.1016/j.jctb.2025.02.001","url":null,"abstract":"<div><div>Esperet and Joret proved that planar graphs with bounded maximum degree are 3-colorable with bounded clustering. Liu and Wood asked whether the conclusion holds with the assumption of the bounded maximum degree replaced by assuming that no two vertices have many common neighbors. We answer this question in positive, in the following stronger form: Let <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>″</mo></mrow></msubsup></math></span> be the complete join of two isolated vertices with a path on <em>t</em> vertices. For any surface Σ, a subgraph-closed class of graphs drawn on Σ is 3-choosable with bounded clustering if and only if there exists <em>t</em> such that <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>″</mo></mrow></msubsup></math></span> does not belong to the class.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 45-67"},"PeriodicalIF":1.2,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394992","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 ,&nbsp;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":"2025-07-01","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}