Journal of Combinatorial Theory Series B最新文献

{"title":"A problem of Erdős and Hajnal on paths with equal-degree endpoints","authors":"Kaizhe Chen ,&nbsp;Jie Ma","doi":"10.1016/j.jctb.2026.01.006","DOIUrl":"10.1016/j.jctb.2026.01.006","url":null,"abstract":"<div><div>We address a problem posed by Erdős and Hajnal in 1991, proving that for all <span><math><mi>n</mi><mo>≥</mo><mn>600</mn></math></span>, every <span><math><mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-vertex graph with at least <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>n</mi><mo>+</mo><mn>1</mn></math></span> edges contains two vertices of equal degree connected by a path of length three. The complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> demonstrates that this edge bound is sharp. We further establish an analogous result for graphs with even order and investigate several related extremal problems.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"179 ","pages":"Pages 1-18"},"PeriodicalIF":1.2,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146134713","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":"Reduced bandwidth: A qualitative strengthening of twin-width in minor-closed classes (and beyond)","authors":"Édouard Bonnet ,&nbsp;O-joung Kwon ,&nbsp;David R. Wood","doi":"10.1016/j.jctb.2025.11.010","DOIUrl":"10.1016/j.jctb.2025.11.010","url":null,"abstract":"<div><div>In a <em>reduction sequence</em> of a graph, vertices are successively identified until the graph has one vertex. At each step, when identifying <em>u</em> and <em>v</em>, each edge incident to exactly one of <em>u</em> and <em>v</em> is coloured red. Bonnet, Kim, Thomassé and Watrigant (2022) <span><span>[19]</span></span> defined the <em>twin-width</em> of a graph <em>G</em> to be the minimum integer <em>k</em> such that there is a reduction sequence of <em>G</em> in which every red graph has maximum degree at most <em>k</em>. For any graph parameter <em>f</em>, we define the <em>reduced f</em> of a graph <em>G</em> to be the minimum integer <em>k</em> such that there is a reduction sequence of <em>G</em> in which every red graph has <em>f</em> at most <em>k</em>. Our focus is on graph classes with bounded reduced bandwidth, which implies and is stronger than bounded twin-width (reduced maximum degree). We show that every proper minor-closed class has bounded reduced bandwidth, which is qualitatively stronger than an analogous result of Bonnet et al. for bounded twin-width. In many instances, we also make quantitative improvements. For example, all previous upper bounds on the twin-width of planar graphs were at least 2¹⁰⁰⁰. We show that planar graphs have reduced bandwidth at most 466 and twin-width at most 583. Our bounds for graphs of Euler genus <em>γ</em> are <span><math><mi>O</mi><mo>(</mo><mi>γ</mi><mo>)</mo></math></span>. Lastly, we show that fixed powers of graphs in a proper minor-closed class have bounded reduced bandwidth (irrespective of the degree of the vertices). In particular, we show that map graphs of Euler genus <em>γ</em> have reduced bandwidth <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>γ</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math></span>. Lastly, we separate twin-width and reduced bandwidth by showing that any infinite class of expanders excluding a fixed complete bipartite subgraph has unbounded reduced bandwidth, while there are bounded-degree expanders with twin-width at most 6.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"178 ","pages":"Pages 27-66"},"PeriodicalIF":1.2,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685989","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":"Cycle matroids of graphings: From convergence to duality","authors":"Kristóf Bérczi ,&nbsp;Márton Borbényi ,&nbsp;László Lovász ,&nbsp;László Márton Tóth","doi":"10.1016/j.jctb.2025.12.003","DOIUrl":"10.1016/j.jctb.2025.12.003","url":null,"abstract":"<div><div>A recent line of research has concentrated on exploring the links between analytic and combinatorial theories of submodularity, uncovering several key connections between them. In this context, Lovász initiated the study of matroids from an analytic point of view and introduced the cycle matroid of a graphing <span><span>[23]</span></span>. Motivated by the limit theory of graphs, the authors introduced a form of right-convergence, called quotient-convergence, for a sequence of submodular setfunctions, leading to a notion of convergence for matroids through their rank functions <span><span>[5]</span></span>.</div><div>In this paper, we study the connection between local-global convergence of graphs and quotient-convergence of their cycle matroids. We characterize the exposed points of associated convex sets, forming an analytic counterpart of matroid independence- and base-polytopes. Finally, we consider dual planar graphings and show that the cycle matroid of one is the cocycle matroid of its dual if and only if the underlying graphings are hyperfinite.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"178 ","pages":"Pages 118-144"},"PeriodicalIF":1.2,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145760380","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":"Colorings of k-sets with low discrepancy on small sets","authors":"Pavel Pudlák ,&nbsp;Vojtěch Rödl","doi":"10.1016/j.jctb.2025.12.001","DOIUrl":"10.1016/j.jctb.2025.12.001","url":null,"abstract":"<div><div>For <span><math><mn>0</mn><mo>&lt;</mo><mi>δ</mi><mo>≤</mo><mn>1</mn></math></span>, let <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>m</mi><mo>;</mo><mi>δ</mi><mo>)</mo></math></span> denote the smallest <em>N</em> such that every coloring of <em>k</em>-element subsets by two colors yields an <em>m</em>-element set <em>M</em> with relative discrepancy <em>δ</em>, which means that one color class has at least <span><math><mo>(</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>δ</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>m</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> elements. The number <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>m</mi><mo>;</mo><mi>δ</mi><mo>)</mo></math></span> may be viewed as an extension of the usual <em>k</em>-hypergraph Ramsey number because <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>m</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>m</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. Our main result is the following theorem. <em>For some constants</em> <span><math><mi>c</mi><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span><em>, and</em> <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span><em>, and for all</em> <span><math><mi>k</mi><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span><em>,</em> <span><math><mi>c</mi><mi>log</mi><mo>⁡</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>≤</mo><mi>k</mi><mo>/</mo><mn>11</mn></math></span><em>,</em><span><span><span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>+</mo><mi>n</mi><mo>;</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mi>ε</mi><mi>n</mi></mrow></msup><mo>)</mo><mo>≥</mo><msub><mrow><mi>tw</mi></mrow><mrow><mo>⌊</mo><mi>k</mi><mo>/</mo><mi>n</mi><mo>⌋</mo></mrow></msub><mo>(</mo><mn>2</mn><mo>)</mo><mo>.</mo></math></span></span></span> In particular, for <span><math><mi>n</mi><mo>=</mo><mo>⌈</mo><mi>c</mi><mi>log</mi><mo>⁡</mo><mi>k</mi><mo>⌉</mo></math></span>, we get a tower of height <span><math><mi>δ</mi><mi>k</mi><mo>/</mo><mi>log</mi><mo>⁡</mo><mi>k</mi></math></span> and relative discrepancy polynomial in <em>k</em>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"178 ","pages":"Pages 79-103"},"PeriodicalIF":1.2,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145731498","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":"Strong log-convexity of genus sequences","authors":"Bojan Mohar","doi":"10.1016/j.jctb.2025.12.005","DOIUrl":"10.1016/j.jctb.2025.12.005","url":null,"abstract":"<div><div>For a graph <em>G</em>, and a nonnegative integer <em>g</em>, let <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the number of 2-cell embeddings of <em>G</em> in an orientable surface of genus <em>g</em> (counted up to the combinatorial homeomorphism equivalence). In 1989, Gross, Robbins, and Tucker <span><span>[21]</span></span> proposed a conjecture that the sequence <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mo>…</mo></math></span> is log-concave for every graph <em>G</em>. This conjecture is reminiscent to the Heron–Rota–Welsh Log Concavity Conjecture that was recently resolved in the affirmative by June Huh et al. <span><span>[24]</span></span>, <span><span>[1]</span></span>, except that it is closer to the notion of Δ-matroids than to the usual matroids. In this short paper, we disprove the Log Concavity Conjecture of Gross, Robbins, and Tucker by providing examples that show strong deviation from log-concavity at multiple terms of their genus sequences.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"178 ","pages":"Pages 164-179"},"PeriodicalIF":1.2,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145902636","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":"Inducibility in H-free graphs and inducibility of Turán graphs","authors":"Raphael Yuster","doi":"10.1016/j.jctb.2025.11.006","DOIUrl":"10.1016/j.jctb.2025.11.006","url":null,"abstract":"&lt;div&gt;&lt;div&gt;For graphs &lt;em&gt;F&lt;/em&gt; and &lt;em&gt;H&lt;/em&gt;, let &lt;span&gt;&lt;math&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; denote the inducibility of &lt;em&gt;F&lt;/em&gt; and let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; denote the inducibility of &lt;em&gt;F&lt;/em&gt; over &lt;em&gt;H&lt;/em&gt;-free graphs. We prove that for almost all graphs &lt;em&gt;F&lt;/em&gt; on a given number of vertices, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; attains infinitely many values as &lt;em&gt;k&lt;/em&gt; varies. For complete partite graphs &lt;em&gt;F&lt;/em&gt; (and, more generally, for symmetrizable families of graphs &lt;em&gt;F&lt;/em&gt;), we prove that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, and is attained by a complete &lt;em&gt;ℓ&lt;/em&gt;-partite graphon &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;We determine the part sizes of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; for all &lt;em&gt;k&lt;/em&gt;, whence determine &lt;span&gt;&lt;math&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, whenever &lt;em&gt;F&lt;/em&gt; is the Turán graph on &lt;em&gt;s&lt;/em&gt; vertices and &lt;em&gt;r&lt;/em&gt; parts, for all &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, which was recently proved by Liu, Mubayi, and Reiher for &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. As a corollary, this determines the inducibility of all Turán graphs on at most 14 vertices. Furthermore, since inducibility is invariant under complement, this determines the inducibility of all matchings and, more generally, all graphs with maximum degree 1, of any size. Similarly, this determines the inducibility of all triangle factors, of any size.&lt;/div&gt;&lt;div&gt;For complete partite graphs &lt;em&gt;F&lt;/em&gt; with at most one singleton part, we prove that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; only attains finitely many values as &lt;em&gt;k&lt;/em&gt; varies; in particular, there exists &lt;span&gt;&lt;math&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is attained by some complete &lt;em&gt;t&lt;/em&gt;-partite graphon. This is best possible as it was shown by Liu, Pikhurko, Sharifzadeh, and Staden that this is not necessarily true if there are two singleton parts.&lt;/div&gt;","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"178 ","pages":"Pages 1-26"},"PeriodicalIF":1.2,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145665695","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":"On the size of two minimal linkages","authors":"Koyo Hayashi ,&nbsp;Ken-ichi Kawarabayashi ,&nbsp;Daiki Kobayashi","doi":"10.1016/j.jctb.2025.11.007","DOIUrl":"10.1016/j.jctb.2025.11.007","url":null,"abstract":"<div><div>We prove that for any two positive integers <span><math><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, if <em>G</em> is a graph, <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are vertex-subsets of <em>G</em>, and <em>G</em> is edge-minimal with respect to the condition that for <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></math></span> there are <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> disjoint paths of <em>G</em> between <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, then <em>G</em> contains at most <span><math><mn>12</mn><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> vertices of degree at least three. This bound is optimal up to a constant factor, as a <span><math><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>×</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-grid <em>G</em> shows.</div><div>The degree-3 treewidth sparsifier theorem, proved by Chekuri and Chuzhoy (2015), states that for any graph <em>G</em> of treewidth at least <em>k</em>, there is a subcubic subgraph of <em>G</em> that has treewidth <span><math><mi>Ω</mi><mo>(</mo><mi>k</mi><mo>/</mo><mi>poly</mi><mspace></mspace><mi>log</mi><mo>⁡</mo><mi>k</mi><mo>)</mo></math></span> and contains <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math></span> vertices of degree three. Our result, together with their proof techniques, reduces the bound <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math></span> in this theorem to <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>, solving their conjecture.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"178 ","pages":"Pages 67-78"},"PeriodicalIF":1.2,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145705132","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":"Integral biflow maximization","authors":"Guoli Ding ,&nbsp;Rongchuan Tao ,&nbsp;Mengxi Yang ,&nbsp;Wenan Zang","doi":"10.1016/j.jctb.2025.12.006","DOIUrl":"10.1016/j.jctb.2025.12.006","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> be a graph with four distinguished vertices, two sources <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and two sinks <span><math><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, let <span><math><mi>c</mi><mo>:</mo><mspace></mspace><mi>E</mi><mo>→</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span> be a capacity function, and let <span><math><mi>P</mi></math></span> be the set of all simple paths in <em>G</em> from <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> to <span><math><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> or from <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> to <span><math><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. A <em>biflow</em> (or 2<em>-commodity flow</em>) in <em>G</em> is an assignment <span><math><mi>f</mi><mo>:</mo><mspace></mspace><mi>P</mi><mo>→</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span> such that <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><mi>Q</mi><mo>∈</mo><mi>P</mi></mrow></msub><mspace></mspace><mi>f</mi><mo>(</mo><mi>Q</mi><mo>)</mo><mo>≤</mo><mi>c</mi><mo>(</mo><mi>e</mi><mo>)</mo></math></span> for all <span><math><mi>e</mi><mo>∈</mo><mi>E</mi></math></span>, whose <em>value</em> is defined to be <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>Q</mi><mo>∈</mo><mi>P</mi></mrow></msub><mspace></mspace><mi>f</mi><mo>(</mo><mi>Q</mi><mo>)</mo></math></span>. A <em>bicut</em> in <em>G</em> is a subset <em>K</em> of <em>E</em> that contains at least one edge from each member of <span><math><mi>P</mi></math></span>, whose <em>capacity</em> is <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><mi>K</mi></mrow></msub><mspace></mspace><mi>c</mi><mo>(</mo><mi>e</mi><mo>)</mo></math></span>. In 1977 Seymour characterized, in terms of forbidden structures, all graphs <em>G</em> for which the max-biflow (integral) min-bicut theorem holds true (that is, the maximum value of an integral biflow is equal to the minimum capacity of a bicut for every capacity function <em>c</em>); such a graph <em>G</em> is referred to as a Seymour graph. Nevertheless, his proof is not algorithmic in nature. In this paper we present a combinatorial polynomial-time algorithm for finding maximum integral biflows in Seymour graphs, which relies heavily on a structural description of such graphs.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"178 ","pages":"Pages 180-210"},"PeriodicalIF":1.2,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145925956","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":"The maximum sum of sizes of non-empty pairwise cross-intersecting families","authors":"Yang Huang ,&nbsp;Yuejian Peng ,&nbsp;Jian Wang","doi":"10.1016/j.jctb.2025.12.004","DOIUrl":"10.1016/j.jctb.2025.12.004","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Two families &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; are cross-intersecting if &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mo&gt;∅&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for any &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. We call &lt;em&gt;t&lt;/em&gt; families &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; pairwise cross-intersecting families if &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are cross-intersecting for &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Additionally, if &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mo&gt;∅&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for each &lt;span&gt;&lt;math&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, then we say that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are non-empty pairwise cross-intersecting. Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; be non-empty pairwise cross-intersecting families with &lt;span&gt;&lt;math&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, we determine the maximum of &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mr","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"178 ","pages":"Pages 145-163"},"PeriodicalIF":1.2,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884405","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":"Universality for graphs with bounded density","authors":"Noga Alon ,&nbsp;Natalie Dodson ,&nbsp;Carmen Jackson ,&nbsp;Rose McCarty ,&nbsp;Rajko Nenadov ,&nbsp;Lani Southern","doi":"10.1016/j.jctb.2026.01.004","DOIUrl":"10.1016/j.jctb.2026.01.004","url":null,"abstract":"<div><div>A graph <em>G</em> is <em>universal</em> for a (finite) family <span><math><mi>H</mi></math></span> of graphs if every <span><math><mi>H</mi><mo>∈</mo><mi>H</mi></math></span> is a subgraph of <em>G</em>. For a given family <span><math><mi>H</mi></math></span>, the goal is to determine the smallest number of edges an <span><math><mi>H</mi></math></span>-universal graph can have. With the aim of unifying a number of recent results, we consider a family of graphs with bounded density. In particular, we construct a graph with<span><span><span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mo>−</mo><mn>1</mn><mo>/</mo><mo>(</mo><mo>⌈</mo><mi>d</mi><mo>⌉</mo><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>)</mo></mrow></math></span></span></span> edges which contains every <em>n</em>-vertex graph with density at most <span><math><mi>d</mi><mo>∈</mo><mi>Q</mi></math></span> (<span><math><mi>d</mi><mo>≥</mo><mn>1</mn></math></span>), which is close to a <span><math><mi>Ω</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>d</mi></mrow></msup><mo>)</mo></math></span> lower bound obtained by counting lifts of a carefully chosen (small) graph. When restricting the maximum degree of such graphs to be constant, we obtain near-optimal universality. If we further assume <span><math><mi>d</mi><mo>∈</mo><mi>N</mi></math></span>, we get an asymptotically optimal construction.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"178 ","pages":"Pages 245-266"},"PeriodicalIF":1.2,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146072031","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}