Journal of Combinatorial Theory Series B最新文献

{"title":"Non-linear Hamilton cycles in linear quasirandom and uniformly dense hypergraphs","authors":"Jie Han ,&nbsp;Xichao Shu ,&nbsp;Guanghui Wang","doi":"10.1016/j.jctb.2025.10.006","DOIUrl":"10.1016/j.jctb.2025.10.006","url":null,"abstract":"<div><div>A <em>k</em>-graph <em>H</em> is called <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>μ</mi><mo>)</mo></math></span><em>-dense</em> if for all not necessarily distinct sets <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> we have <span><math><mi>e</mi><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>≥</mo><mi>p</mi><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><mo>⋯</mo><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>|</mo><mo>−</mo><mi>μ</mi><mo>|</mo><mi>V</mi><mo>(</mo><mi>H</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>k</mi></mrow></msup></math></span>. This is believed to be the weakest form of quasirandomness in <em>k</em>-graphs and also known as <em>linear quasirandomness</em>.</div><div>In this paper, we show that for <span><math><mi>ℓ</mi><mo>&lt;</mo><mi>k</mi></math></span> satisfying <span><math><mo>(</mo><mi>k</mi><mo>−</mo><mi>ℓ</mi><mo>)</mo><mo>∤</mo><mi>k</mi></math></span>, <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>μ</mi><mo>)</mo></math></span>-density plus a minimum <span><math><mo>(</mo><mi>ℓ</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-degree of <span><math><mi>α</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>k</mi><mo>−</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> guarantees Hamilton <em>ℓ</em>-cycles, but requiring a minimum <em>ℓ</em>-degree of <span><math><mi>Ω</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>k</mi><mo>−</mo><mi>ℓ</mi></mrow></msup><mo>)</mo></math></span> instead is not sufficient. This answers a question of Lenz–Mubayi–Mycroft and characterizes the triples <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> when <span><math><mi>k</mi><mo>−</mo><mi>ℓ</mi><mo>∤</mo><mi>k</mi></math></span> such that degenerate choices of <em>p</em> and <em>α</em> force <em>ℓ</em>-Hamiltonicity.</div><div>We actually prove a general result on <em>ℓ</em>-Hamiltonicity in quasirandom <em>k</em>-graphs, assuming a minimum vertex degree and essentially that every two <em>ℓ</em>-sets can be connected by a constant length <em>ℓ</em>-path. This reduces the <em>ℓ</em>-Hamiltonicity problem to the study of the connection property which also allows us to deduce a <span><math><mo>(</mo><mi>k</mi><mo>/</mo><mn>2</mn><mo>)</mo></math></span>-Hamiltonicity result in uniformly dense <em>k</em>-graphs (for even <span><math><mi>k</mi><mo>≥</mo><mn>4</mn></math></span>).</div><div>Our proof uses the lattice-based absorption method in the non-standard way and is the first one that embeds a <em>non-linear</em> Hamilton cycle in linear quasirandom <em>k</em>-graphs.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"177 ","pages":"Pages 1-30"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145442022","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":"When recursion is better than iteration: A linear-time algorithm for directed acyclicity with few error vertices","authors":"Daniel Lokshtanov ,&nbsp;M.S. Ramanujan ,&nbsp;Saket Saurabh","doi":"10.1016/j.jctb.2025.11.002","DOIUrl":"10.1016/j.jctb.2025.11.002","url":null,"abstract":"<div><div>Planarity, bipartiteness and (directed) acyclicity are basic graph properties with classic linear-time recognition algorithms. However, the problems of testing whether a given graph has <em>k</em> vertices whose deletion makes it planar, bipartite or a directed acyclic graph (DAG) are all fundamental NP-complete problems when <em>k</em> is part of the input. As a result, a significant amount of research has been devoted to understanding whether, for every <em>fixed k</em>, these problems admit a polynomial-time algorithm (where the exponent in the polynomial is independent of <em>k</em>) and in particular, whether they admit linear-time algorithms.</div><div>While we now know that for every fixed <em>k</em>, we can test in linear time whether a graph is <em>k</em> vertices away from being planar or bipartite, the best known algorithms in the case of directed acyclicity are the algorithm of Garey and Tarjan [IPL 1978], which runs in time <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><mi>m</mi><mo>)</mo></math></span> and the algorithm of Chen, Liu, Lu, O'Sullivan and Razgon [JACM 2008], which runs in time <span><math><mi>O</mi><mo>(</mo><mi>k</mi><mo>!</mo><mo>⋅</mo><msup><mrow><mn>4</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>⋅</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>⋅</mo><mi>n</mi><mo>⋅</mo><mi>m</mi><mo>)</mo></math></span>, where <em>n</em> and <em>m</em> are the number of vertices and arcs in the input digraph, respectively. In other words, it has remained open whether it is possible to recognize in linear time, a graph that is <em>two vertices</em> away from being acyclic.</div><div>In this paper, we settle this question by giving an algorithm that decides whether a given graph is <em>k</em> vertices away from being acyclic, in time <span><math><mi>O</mi><mo>(</mo><mi>k</mi><mo>!</mo><mo>⋅</mo><msup><mrow><mn>4</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>⋅</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>5</mn></mrow></msup><mo>⋅</mo><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo><mo>)</mo></math></span>. That is, for every fixed <em>k</em>, our algorithm runs in time <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></math></span>, thus mirroring the case for planarity and bipartiteness.</div><div>We obtain our algorithm by introducing a general methodology that shaves off a factor of <em>n</em> from certain algorithms that use the powerful technique of <em>iterative compression</em>. The two main features of our methodology are: (i) This is the first generic technique for designing linear-time <span>FPT</span> algorithms for <em>directed cut problems</em> and (ii) it can be used in combination with future improvements in algorithms for the so-called <em>compression</em> version of other well-studied cut problems such as <span>Multicut</span> and <span>Directed Subset Feedback Vertex Set</span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"177 ","pages":"Pages 143-185"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145546295","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":"Extremal density for subdivisions with length or sparsity constraints","authors":"Jaehoon Kim ,&nbsp;Hong Liu ,&nbsp;Yantao Tang ,&nbsp;Guanghui Wang ,&nbsp;Donglei Yang ,&nbsp;Fan Yang","doi":"10.1016/j.jctb.2025.11.001","DOIUrl":"10.1016/j.jctb.2025.11.001","url":null,"abstract":"<div><div>Given a graph <em>H</em>, a balanced subdivision of <em>H</em> is obtained by replacing all edges of <em>H</em> with internally disjoint paths of the same length. In this paper, we prove that for any graph <em>H</em>, a linear-in-<span><math><mi>e</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> bound on average degree guarantees a balanced <em>H</em>-subdivision. This strengthens an old result of Bollobás and Thomason, and resolves a question of Gil-Fernández, Hyde, Liu, Pikhurko and Wu.</div><div>We observe that this linear bound on average degree is best possible whenever <em>H</em> is logarithmically dense. We further show that this logarithmic density is the critical threshold: for many graphs <em>H</em> below this density, its subdivisions are forcible by a sublinear-in-<span><math><mi>e</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> bound on average degree. We provide such examples by proving that the subdivisions of any almost bipartite graph <em>H</em> with sublogarithmic density are forcible by a sublinear-in-<span><math><mi>e</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> bound on average degree, provided that <em>H</em> satisfies some additional separability condition.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"177 ","pages":"Pages 67-104"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145546296","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":"Reuniting χ-boundedness with polynomial χ-boundedness","authors":"Maria Chudnovsky ,&nbsp;Linda Cook ,&nbsp;James Davies ,&nbsp;Sang-il Oum","doi":"10.1016/j.jctb.2025.08.002","DOIUrl":"10.1016/j.jctb.2025.08.002","url":null,"abstract":"<div><div>A class <span><math><mi>F</mi></math></span> of graphs is <em>χ</em>-bounded if there is a function <em>f</em> such that <span><math><mi>χ</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mi>f</mi><mo>(</mo><mi>ω</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>)</mo></math></span> for all induced subgraphs <em>H</em> of a graph in <span><math><mi>F</mi></math></span>. If <em>f</em> can be chosen to be a polynomial, we say that <span><math><mi>F</mi></math></span> is polynomially <em>χ</em>-bounded. Esperet proposed a conjecture that every <em>χ</em>-bounded class of graphs is polynomially <em>χ</em>-bounded. This conjecture has been disproved; it has been shown that there are classes of graphs that are <em>χ</em>-bounded but not polynomially <em>χ</em>-bounded. Nevertheless, inspired by Esperet's conjecture, we introduce Pollyanna classes of graphs. A class <span><math><mi>C</mi></math></span> of graphs is Pollyanna if <span><math><mi>C</mi><mo>∩</mo><mi>F</mi></math></span> is polynomially <em>χ</em>-bounded for every <em>χ</em>-bounded class <span><math><mi>F</mi></math></span> of graphs. We prove that several classes of graphs are Pollyanna and also present some proper classes of graphs that are not Pollyanna.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 30-73"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144907692","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":"Splitting-off in hypergraphs","authors":"Kristóf Bérczi ,&nbsp;Karthekeyan Chandrasekaran ,&nbsp;Tamás Király ,&nbsp;Shubhang Kulkarni","doi":"10.1016/j.jctb.2025.09.004","DOIUrl":"10.1016/j.jctb.2025.09.004","url":null,"abstract":"<div><div>The splitting-off operation in undirected graphs is a fundamental reduction operation that detaches all edges incident to a given vertex and adds new edges between the neighbors of that vertex while preserving their degrees. Lovász <span><span>[47]</span></span>, <span><span>[49]</span></span> and Mader <span><span>[50]</span></span> showed the existence of this operation while preserving global and local connectivities respectively in graphs under certain conditions. These results have far-reaching applications in graph algorithms literature <span><span>[3]</span></span>, <span><span>[9]</span></span>, <span><span>[10]</span></span>, <span><span>[14]</span></span>, <span><span>[19]</span></span>, <span><span>[24]</span></span>, <span><span>[25]</span></span>, <span><span>[26]</span></span>, <span><span>[27]</span></span>, <span><span>[28]</span></span>, <span><span>[31]</span></span>, <span><span>[32]</span></span>, <span><span>[34]</span></span>, <span><span>[35]</span></span>, <span><span>[37]</span></span>, <span><span>[40]</span></span>, <span><span>[42]</span></span>, <span><span>[43]</span></span>, <span><span>[48]</span></span>, <span><span>[50]</span></span>, <span><span>[51]</span></span>, <span><span>[52]</span></span>, <span><span>[53]</span></span>. In this work, we introduce a splitting-off operation in hypergraphs. We show that there exists a local connectivity preserving complete splitting-off in hypergraphs and give a strongly polynomial-time algorithm to compute it in weighted hypergraphs. We illustrate the usefulness of our splitting-off operation in hypergraphs by showing two applications: (1) we give a constructive characterization of <em>k</em>-hyperedge-connected hypergraphs and (2) we give an alternate proof of an approximate min-max relation for max Steiner rooted-connected orientation of graphs and hypergraphs (due to Király and Lau (2008) <span><span>[40]</span></span>). Our proof of the approximate min-max relation for graphs circumvents the Nash-Williams' strong orientation theorem and uses tools developed for hypergraphs.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 319-383"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145265869","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":"Asymptotic half-grid and full-grid minors","authors":"Sandra Albrechtsen,&nbsp;Matthias Hamann","doi":"10.1016/j.jctb.2025.10.003","DOIUrl":"10.1016/j.jctb.2025.10.003","url":null,"abstract":"<div><div>We prove that every locally finite, quasi-transitive graph with a thick end whose cycle space is generated by cycles of bounded length contains the full-grid as an asymptotic minor and as a diverging minor. This in particular includes all locally finite Cayley graphs of finitely presented groups that are not virtually free, and partially solves problems of Georgakopoulos and Papasoglu and of Georgakopoulos and Hamann.</div><div>Additionally, we show that every (not necessarily quasi-transitive) graph of finite maximum degree which has a thick end and whose cycle space is generated by cycles of bounded length contains the half-grid as an asymptotic minor and as a diverging minor.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 440-485"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145361715","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":"Connectoids I: A universal end space theory","authors":"Nathan Bowler,&nbsp;Florian Reich","doi":"10.1016/j.jctb.2025.06.003","DOIUrl":"10.1016/j.jctb.2025.06.003","url":null,"abstract":"<div><div>In this series we introduce and investigate the concept of <em>connectoids</em>, which captures the connectivity structure of various discrete objects like undirected graphs, directed graphs, bidirected graphs, hypergraphs or finitary matroids.</div><div>In this paper we develop a universal end space theory based on connectoids: the end spaces of connectoids unify the existing end spaces of undirected and directed graphs and establish end spaces for bidirected graphs, hypergraphs and finitary matroids.</div><div>The main result shows that the tangle-like description of ends in undirected graphs, called <em>directions</em>, extends to connectoids: there is a one-to-one correspondence between the “directions” of a connectoid and its ends. Furthermore, we generalise normal trees of undirected graphs to connectoids and show that normal trees represent the ends of a connectoid as they do for undirected graphs.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 584-606"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144515440","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":"An infinite family of simple graphs underlying chiral, orientable reflexible and non-orientable rotary maps","authors":"Isabel Hubard ,&nbsp;Primož Potočnik ,&nbsp;Primož Šparl","doi":"10.1016/j.jctb.2025.10.001","DOIUrl":"10.1016/j.jctb.2025.10.001","url":null,"abstract":"<div><div>In this paper, we provide the first known infinite family of simple graphs, each of which is the skeleton of a chiral map, a skeleton of a reflexible map on an orientable surface, as well as a skeleton of a reflexible map on a non-orientable surface. This family consists of all lexicographic products <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>[</mo><mi>m</mi><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>]</mo></math></span>, where <span><math><mi>m</mi><mo>≥</mo><mn>3</mn></math></span>, <span><math><mi>n</mi><mo>=</mo><mi>s</mi><mi>m</mi></math></span>, with <em>s</em> an integer not divisible by 4. This answers a question posed by Wilson in 2002.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 384-403"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145332383","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":"Transversals via regularity","authors":"Yangyang Cheng ,&nbsp;Katherine Staden","doi":"10.1016/j.jctb.2025.10.004","DOIUrl":"10.1016/j.jctb.2025.10.004","url":null,"abstract":"<div><div>Given graphs <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> all on the same vertex set and a graph <em>H</em> with <span><math><mi>e</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mi>s</mi></math></span>, a copy of <em>H</em> is <em>transversal</em> or <em>rainbow</em> if it contains at most one edge from each <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span>. We study the case when <em>H</em> is spanning and explore how the regularity blow-up method, that has been so successful in the uncoloured setting, can be used to find transversals. We provide the analogues of the tools required to apply this method in the transversal setting. Our main result is a blow-up lemma for transversals that applies to separable bounded degree graphs <em>H</em>.</div><div>Our proofs use weak regularity in the 3-uniform hypergraph whose edges are those <em>xyc</em> where <em>xy</em> is an edge in the graph <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span>. We apply our lemma to give a large class of spanning 3-uniform linear hypergraphs <em>H</em> such that any sufficiently large uniformly dense <em>n</em>-vertex 3-uniform hypergraph with minimum vertex degree <span><math><mi>Ω</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> contains <em>H</em> as a subhypergraph. This extends work of Lenz, Mubayi and Mycroft.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 498-550"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145383220","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":"Thomassen's theorem on the two-linkage problem in acyclic digraphs: A shorter proof","authors":"Paul Seymour","doi":"10.1016/j.jctb.2025.08.006","DOIUrl":"10.1016/j.jctb.2025.08.006","url":null,"abstract":"<div><div>Let <em>G</em> be an acyclic digraph, and let <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, where <span><math><mi>a</mi><mo>,</mo><mi>b</mi></math></span> are sources, <span><math><mi>c</mi><mo>,</mo><mi>d</mi></math></span> are sinks, and every other vertex has in-degree and out-degree at least two. In 1985, Thomassen showed that there do not exist disjoint directed paths from <em>a</em> to <em>c</em> and from <em>b</em> to <em>d</em>, if and only if <em>G</em> can be drawn in a closed disc with <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi></math></span> drawn in the boundary in order. We give a shorter proof.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 97-100"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049038","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}