Journal of Combinatorial Theory Series B最新文献

{"title":"Excluded minors for the Klein bottle II. Cascades","authors":"Bojan Mohar ,&nbsp;Petr Škoda","doi":"10.1016/j.jctb.2023.12.006","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.12.006","url":null,"abstract":"<div><p>Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I, it was shown that graphs that are critical for embeddings into surfaces of Euler genus <em>k</em><span> or for embeddings into nonorientable surface of genus </span><em>k</em><span><span> are built from 3-connected components, called hoppers and cascades. In Part II, all cascades for Euler genus 2 are classified. As a consequence, the complete list of obstructions of connectivity 2 for embedding graphs into the </span>Klein bottle is obtained.</span></p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139433917","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 graphs without long induced paths","authors":"Oscar Defrain ,&nbsp;Jean-Florent Raymond","doi":"10.1016/j.jctb.2023.12.003","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.12.003","url":null,"abstract":"<div><p>Graphs of bounded degeneracy are known to contain induced paths of order <span><math><mi>Ω</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> when they contain a path of order <em>n</em>, as proved by Nešetřil and Ossona de Mendez (2012). In 2016 Esperet, Lemoine, and Maffray conjectured that this bound could be improved to <span><math><mi>Ω</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>c</mi></mrow></msup><mo>)</mo></math></span> for some constant <span><math><mi>c</mi><mo>&gt;</mo><mn>0</mn></math></span> depending on the degeneracy.</p><p>We disprove this conjecture by constructing, for arbitrarily large values of <em>n</em>, a graph that is 2-degenerate, has a path of order <em>n</em>, and where all induced paths have order <span><math><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>. We also show that the graphs we construct have linearly bounded coloring numbers.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139107742","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":"Count and cofactor matroids of highly connected graphs","authors":"Dániel Garamvölgyi ,&nbsp;Tibor Jordán ,&nbsp;Csaba Király","doi":"10.1016/j.jctb.2023.12.004","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.12.004","url":null,"abstract":"<div><p>We consider two types of matroids defined on the edge set of a graph <em>G</em>: count matroids <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, in which independence is defined by a sparsity count involving the parameters <em>k</em> and <em>ℓ</em>, and the <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-cofactor matroid <span><math><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, in which independence is defined by linear independence in the cofactor matrix of <em>G</em>. We show, for each pair <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>ℓ</mi><mo>)</mo></math></span>, that if <em>G</em> is sufficiently highly connected, then <span><math><mi>G</mi><mo>−</mo><mi>e</mi></math></span> has maximum rank for all <span><math><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, and the matroid <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is connected. These results unify and extend several previous results, including theorems of Nash-Williams and Tutte (<span><math><mi>k</mi><mo>=</mo><mi>ℓ</mi><mo>=</mo><mn>1</mn></math></span>), and Lovász and Yemini (<span><math><mi>k</mi><mo>=</mo><mn>2</mn><mo>,</mo><mi>ℓ</mi><mo>=</mo><mn>3</mn></math></span>). We also prove that if <em>G</em> is highly connected, then the vertical connectivity of <span><math><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is also high.</p><p>We use these results to generalize Whitney's celebrated result on the graphic matroid of <em>G</em> (which corresponds to <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>) to all count matroids and to the <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-cofactor matroid: if <em>G</em> is highly connected, depending on <em>k</em> and <em>ℓ</em>, then the count matroid <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> uniquely determines <em>G</em>; and similarly, if <em>G</em> is 14-connected, then its <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-cofactor matroid <span><math><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> uniquely determines <em>G</em>. We also derive similar results for the <em>t</em>-fold union of the <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-cofactor matroid, and use them to prove that every 24-connected graph has a spanning tree <em>T</em> for which <span><math><mi>G</mi><mo>−</mo><mi>E</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> is 3-connected, whi","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895623001120/pdfft?md5=3aa4475308b3f1d90b43521f41db45ba&pid=1-s2.0-S0095895623001120-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139107741","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":"Turán graphs with bounded matching number","authors":"Noga Alon ,&nbsp;Péter Frankl","doi":"10.1016/j.jctb.2023.12.002","DOIUrl":"10.1016/j.jctb.2023.12.002","url":null,"abstract":"<div><p><span>We determine the maximum possible number of edges of a graph with </span><em>n</em><span> vertices, matching number at most </span><em>s</em> and clique number at most <em>k</em> for all admissible values of the parameters.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138634766","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 a problem of El-Zahar and Erdős","authors":"Tung Nguyen ,&nbsp;Alex Scott ,&nbsp;Paul Seymour","doi":"10.1016/j.jctb.2023.11.004","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.11.004","url":null,"abstract":"<div><p>Two subgraphs <span><math><mi>A</mi><mo>,</mo><mi>B</mi></math></span> of a graph <em>G</em> are <em>anticomplete</em> if they are vertex-disjoint and there are no edges joining them. Is it true that if <em>G</em><span> is a graph with bounded clique number, and sufficiently large chromatic number, then it has two anticomplete subgraphs, both with large chromatic number? This is a question raised by El-Zahar and Erdős in 1986, and remains open. If so, then at least there should be two anticomplete subgraphs both with large minimum degree, and that is one of our results.</span></p><p>We prove two variants of this. First, a strengthening: we can ask for one of the two subgraphs to have large chromatic number: that is, for all <span><math><mi>t</mi><mo>,</mo><mi>c</mi><mo>≥</mo><mn>1</mn></math></span> there exists <span><math><mi>d</mi><mo>≥</mo><mn>1</mn></math></span> such that if <em>G</em> has chromatic number at least <em>d</em>, and does not contain the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> as a subgraph, then there are anticomplete subgraphs <span><math><mi>A</mi><mo>,</mo><mi>B</mi></math></span>, where <em>A</em> has minimum degree at least <em>c</em> and <em>B</em> has chromatic number at least <em>c</em>.</p><p>Second, we look at what happens if we replace the hypothesis that <em>G</em> has sufficiently large chromatic number with the hypothesis that <em>G</em> has sufficiently large minimum degree. This, together with excluding <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>, is <em>not</em> enough to guarantee two anticomplete subgraphs both with large minimum degree; but it works if instead of excluding <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span><span> we exclude the complete bipartite graph </span><span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span>. More exactly: for all <span><math><mi>t</mi><mo>,</mo><mi>c</mi><mo>≥</mo><mn>1</mn></math></span> there exists <span><math><mi>d</mi><mo>≥</mo><mn>1</mn></math></span> such that if <em>G</em> has minimum degree at least <em>d</em>, and does not contain the complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span> as a subgraph, then there are two anticomplete subgraphs both with minimum degree at least <em>c</em>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138570098","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":"Graph partitions under average degree constraint","authors":"Yan Wang ,&nbsp;Hehui Wu","doi":"10.1016/j.jctb.2023.11.006","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.11.006","url":null,"abstract":"<div><p>In this paper, we prove that every graph with average degree at least <span><math><mi>s</mi><mo>+</mo><mi>t</mi><mo>+</mo><mn>2</mn></math></span> has a vertex partition into two parts, such that one part has average degree at least <em>s</em>, and the other part has average degree at least <em>t</em>. This solves a conjecture of Csóka, Lo, Norin, Wu and Yepremyan.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138484338","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":"Hitting all maximum stable sets in P5-free graphs","authors":"Sepehr Hajebi ,&nbsp;Yanjia Li ,&nbsp;Sophie Spirkl","doi":"10.1016/j.jctb.2023.11.005","DOIUrl":"10.1016/j.jctb.2023.11.005","url":null,"abstract":"<div><p>We prove that every <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span><span>-free graph of bounded clique number contains a small hitting set of all its maximum stable sets (where </span><span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> denotes the <em>t</em>-vertex path, and for graphs <span><math><mi>G</mi><mo>,</mo><mi>H</mi></math></span>, we say <em>G</em> is <em>H-free</em><span> if no induced subgraph of </span><em>G</em> is isomorphic to <em>H</em>).</p><p>More generally, let us say a class <span><math><mi>C</mi></math></span> of graphs is <em>η-bounded</em> if there exists a function <span><math><mi>h</mi><mo>:</mo><mi>N</mi><mo>→</mo><mi>N</mi></math></span> such that <span><math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>h</mi><mo>(</mo><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> for every graph <span><math><mi>G</mi><mo>∈</mo><mi>C</mi></math></span>, where <span><math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> denotes smallest cardinality of a hitting set of all maximum stable sets in <em>G</em>, and <span><math><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the clique number of <em>G</em>. Also, <span><math><mi>C</mi></math></span> is said to be <em>polynomially η-bounded</em> if in addition <em>h</em> can be chosen to be a polynomial.</p><p>We introduce <em>η</em>-boundedness inspired by a question of Alon (asking how large <span><math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> can be for a 3-colourable graph <em>G</em>), and motivated by a number of meaningful similarities to <em>χ</em>-boundedness, namely,</p><ul><li><span>•</span><span><p>given a graph <em>G</em>, we have <span><math><mi>η</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mi>ω</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> for every induced subgraph <em>H</em> of <em>G</em> if and only if <em>G</em> is perfect;</p></span></li><li><span>•</span><span><p>there are graphs <em>G</em> with both <span><math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and the girth of <em>G</em> arbitrarily large; and</p></span></li><li><span>•</span><span><p>if <span><math><mi>C</mi></math></span> is a hereditary class of graphs which is polynomially <em>η</em>-bounded, then <span><math><mi>C</mi></math></span> satisfies the Erdős-Hajnal conjecture.</p></span></li></ul> The second bullet above in particular suggests an analogue of the Gyárfás-Sumner conjecture, that the class of all <em>H</em>-free graphs is <em>η</em>-bounded if (and only if) <em>H</em> is a forest. Like <em>χ</em>-boundedness, the case where <em>H</em> is a star is easy to verify, and we prove two non-trivial extensions of this: <em>H</em>-free graphs are <em>η</em>-bounded if (1) <em>H</em> has a vertex incident with all edges of <em>H</em>, or (2) <em>H</em> can be obtained from a star by subdividing at most one edge, exactly once.<p>Unlike <em>χ</em>-boundedness","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138455110","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":"Dimension is polynomial in height for posets with planar cover graphs","authors":"Jakub Kozik ,&nbsp;Piotr Micek ,&nbsp;William T. Trotter","doi":"10.1016/j.jctb.2023.10.009","DOIUrl":"10.1016/j.jctb.2023.10.009","url":null,"abstract":"<div><p>We show that height <em>h</em><span> posets that have planar cover graphs have dimension </span><span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math></span>. Previously, the best upper bound was <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></msup></math></span><span>. Planarity plays a key role in our arguments, since there are posets such that (1) dimension is exponential in height and (2) the cover graph excludes </span><span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> as a minor.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138455878","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":"Dirac-type conditions for spanning bounded-degree hypertrees","authors":"Matías Pavez-Signé ,&nbsp;Nicolás Sanhueza-Matamala ,&nbsp;Maya Stein","doi":"10.1016/j.jctb.2023.11.002","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.11.002","url":null,"abstract":"<div><p>We prove that for fixed <em>k</em>, every <em>k</em><span>-uniform hypergraph on </span><em>n</em> vertices and of minimum codegree at least <span><math><mi>n</mi><mo>/</mo><mn>2</mn><mo>+</mo><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> contains every spanning tight <em>k</em>-tree of bounded vertex degree as a subgraph. This generalises a well-known result of Komlós, Sárközy and Szemerédi for graphs. Our result is asymptotically sharp. We also prove an extension of our result to hypergraphs that satisfy some weak quasirandomness conditions.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138430649","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":"Edge-colouring graphs with local list sizes","authors":"Marthe Bonamy ,&nbsp;Michelle Delcourt ,&nbsp;Richard Lang ,&nbsp;Luke Postle","doi":"10.1016/j.jctb.2023.10.010","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.10.010","url":null,"abstract":"<div><p>The famous List Colouring Conjecture from the 1970s states that for every graph <em>G</em> the chromatic index of <em>G</em><span> is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph </span><em>G</em><span> with sufficiently large maximum degree Δ and minimum degree </span><span><math><mi>δ</mi><mo>≥</mo><msup><mrow><mi>ln</mi></mrow><mrow><mn>25</mn></mrow></msup><mo>⁡</mo><mi>Δ</mi></math></span>, the following holds: for every assignment <em>L</em> of lists of colours to the edges of <em>G</em>, such that <span><math><mo>|</mo><mi>L</mi><mo>(</mo><mi>e</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mo>⋅</mo><mi>max</mi><mo>⁡</mo><mrow><mo>{</mo><mi>deg</mi><mo>⁡</mo><mo>(</mo><mi>u</mi><mo>)</mo><mo>,</mo><mi>deg</mi><mo>⁡</mo><mo>(</mo><mi>v</mi><mo>)</mo><mo>}</mo></mrow></math></span> for each edge <span><math><mi>e</mi><mo>=</mo><mi>u</mi><mi>v</mi></math></span>, there is an <em>L</em>-edge-colouring of <em>G</em>. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, <em>k</em><span>-uniform hypergraphs, and recently Molloy generalized Kahn's original result to correspondence colouring as well as its hypergraph generalization. We prove local versions of all of these generalizations by showing a weighted version that simultaneously implies all of our results.</span></p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138430648","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}