Journal of Combinatorial Theory Series B最新文献

{"title":"Tight bounds for divisible subdivisions","authors":"Shagnik Das ,&nbsp;Nemanja Draganić ,&nbsp;Raphael Steiner","doi":"10.1016/j.jctb.2023.10.011","DOIUrl":"10.1016/j.jctb.2023.10.011","url":null,"abstract":"<div><p>Alon and Krivelevich proved that for every <em>n</em>-vertex subcubic graph <em>H</em> and every integer <span><math><mi>q</mi><mo>≥</mo><mn>2</mn></math></span> there exists a (smallest) integer <span><math><mi>f</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>H</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> such that every <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span>-minor contains a subdivision of <em>H</em> in which the length of every subdivision-path is divisible by <em>q</em>. Improving their superexponential bound, we show that <span><math><mi>f</mi><mo>(</mo><mi>H</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>21</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>q</mi><mi>n</mi><mo>+</mo><mn>8</mn><mi>n</mi><mo>+</mo><mn>14</mn><mi>q</mi></math></span>, which is optimal up to a constant multiplicative factor.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"165 ","pages":"Pages 1-19"},"PeriodicalIF":1.4,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895623000941/pdfft?md5=0b6cb15d113f5a221914b5ec07224f3a&pid=1-s2.0-S0095895623000941-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138289379","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":"Treewidth versus clique number. II. Tree-independence number","authors":"Clément Dallard ,&nbsp;Martin Milanič ,&nbsp;Kenny Štorgel","doi":"10.1016/j.jctb.2023.10.006","DOIUrl":"10.1016/j.jctb.2023.10.006","url":null,"abstract":"<div><p>In 2020, we initiated a systematic study of graph classes in which the treewidth can only be large due to the presence of a large clique, which we call <span><math><mo>(</mo><mrow><mi>tw</mi></mrow><mo>,</mo><mi>ω</mi><mo>)</mo></math></span>-bounded. The family of <span><math><mo>(</mo><mrow><mi>tw</mi></mrow><mo>,</mo><mi>ω</mi><mo>)</mo></math></span>-bounded graph classes provides a unifying framework for a variety of very different families of graph classes, including graph classes of bounded treewidth, graph classes of bounded independence number, intersection graphs of connected subgraphs of graphs with bounded treewidth, and graphs in which all minimal separators are of bounded size. While Chaplick and Zeman showed in 2017 that <span><math><mo>(</mo><mrow><mi>tw</mi></mrow><mo>,</mo><mi>ω</mi><mo>)</mo></math></span>-bounded graph classes enjoy some good algorithmic properties related to clique and coloring problems, it is an interesting open problem to which extent <span><math><mo>(</mo><mrow><mi>tw</mi></mrow><mo>,</mo><mi>ω</mi><mo>)</mo></math></span>-boundedness has useful algorithmic implications for problems related to independent sets. We provide a partial answer to this question by identifying a sufficient condition for <span><math><mo>(</mo><mrow><mi>tw</mi></mrow><mo>,</mo><mi>ω</mi><mo>)</mo></math></span>-bounded graph classes to admit a polynomial-time algorithm for the Maximum Weight Independent Packing problem and, as a consequence, for the weighted variants of the Independent Set and Induced Matching problems.</p><p>Our approach is based on a new min-max graph parameter related to tree decompositions. We define the <em>independence number</em> of a tree decomposition <span><math><mi>T</mi></math></span> of a graph as the maximum independence number over all subgraphs of <em>G</em> induced by some bag of <span><math><mi>T</mi></math></span>. The <em>tree-independence number</em> of a graph <em>G</em> is then defined as the minimum independence number over all tree decompositions of <em>G</em>. Boundedness of the tree-independence number is a refinement of <span><math><mo>(</mo><mrow><mi>tw</mi></mrow><mo>,</mo><mi>ω</mi><mo>)</mo></math></span>-boundedness that is still general enough to hold for all the aforementioned families of graph classes. Generalizing a result on chordal graphs due to Cameron and Hell from 2006, we show that if a graph is given together with a tree decomposition with bounded independence number, then the Maximum Weight Independent Packing problem can be solved in polynomial time. Applications of our general algorithmic result to specific graph classes are given in the third paper of the series [Dallard, Milanič, and Štorgel, Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure].</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"164 ","pages":"Pages 404-442"},"PeriodicalIF":1.4,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895623000886/pdfft?md5=b1bea8202446f9b5f80995ccca2f2480&pid=1-s2.0-S0095895623000886-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72364894","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":"The immersion-minimal infinitely edge-connected graph","authors":"Paul Knappe ,&nbsp;Jan Kurkofka","doi":"10.1016/j.jctb.2023.10.007","DOIUrl":"10.1016/j.jctb.2023.10.007","url":null,"abstract":"<div><p>We show that there is a unique immersion-minimal infinitely edge-connected graph: every such graph contains the halved Farey graph, which is itself infinitely edge-connected, as an immersion minor.</p><p>By contrast, any minimal list of infinitely edge-connected graphs represented in all such graphs as topological minors must be uncountable.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"164 ","pages":"Pages 492-516"},"PeriodicalIF":1.4,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72364891","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":"Polynomial bounds for chromatic number. V. Excluding a tree of radius two and a complete multipartite graph","authors":"Alex Scott ,&nbsp;Paul Seymour","doi":"10.1016/j.jctb.2023.10.004","DOIUrl":"10.1016/j.jctb.2023.10.004","url":null,"abstract":"<div><p>The Gyárfás-Sumner conjecture says that for every forest <em>H</em> and every integer <em>k</em>, if <em>G</em> is <em>H</em>-free and does not contain a clique on <em>k</em> vertices then it has bounded chromatic number. (A graph is <em>H-free</em> if it does not contain an induced copy of <em>H</em>.) Kierstead and Penrice proved it for trees of radius at most two, but otherwise the conjecture is known only for a few simple types of forest. More is known if we exclude a complete bipartite subgraph instead of a clique: Rödl showed that, for every forest <em>H</em>, if <em>G</em> is <em>H</em>-free and does not contain <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 it has bounded chromatic number. In an earlier paper with Sophie Spirkl, we strengthened Rödl's result, showing that for every forest <em>H</em>, the bound on chromatic number can be taken to be polynomial in <em>t</em>. In this paper, we prove a related strengthening of the Kierstead-Penrice theorem, showing that for every tree <em>H</em> of radius two and integer <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>, if <em>G</em> is <em>H</em>-free and does not contain as a subgraph the complete <em>d</em>-partite graph with parts of cardinality <em>t</em>, then its chromatic number is at most polynomial in <em>t</em>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"164 ","pages":"Pages 473-491"},"PeriodicalIF":1.4,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72364895","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":"Induced subgraphs and tree decompositions VII. Basic obstructions in H-free graphs","authors":"Tara Abrishami ,&nbsp;Bogdan Alecu ,&nbsp;Maria Chudnovsky ,&nbsp;Sepehr Hajebi ,&nbsp;Sophie Spirkl","doi":"10.1016/j.jctb.2023.10.008","DOIUrl":"10.1016/j.jctb.2023.10.008","url":null,"abstract":"<div><p>We say a class <span><math><mi>C</mi></math></span> of graphs is <em>clean</em> if for every positive integer <em>t</em> there exists a positive integer <span><math><mi>w</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> such that every graph in <span><math><mi>C</mi></math></span> with treewidth more than <span><math><mi>w</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> contains an induced subgraph isomorphic to one of the following: the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>, 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>, a subdivision of the <span><math><mo>(</mo><mi>t</mi><mo>×</mo><mi>t</mi><mo>)</mo></math></span>-wall or the line graph of a subdivision of the <span><math><mo>(</mo><mi>t</mi><mo>×</mo><mi>t</mi><mo>)</mo></math></span>-wall. In this paper, we adapt a method due to Lozin and Razgon (building on earlier ideas of Weißauer) to prove that the class of all <em>H-free</em> graphs (that is, graphs with no induced subgraph isomorphic to a fixed graph <em>H</em>) is clean if and only if <em>H</em> is a forest whose components are subdivided stars.</p><p>Their method is readily applied to yield the above characterization. However, our main result is much stronger: for every forest <em>H</em> as above, we show that forbidding certain connected graphs containing <em>H</em> as an induced subgraph (rather than <em>H</em> itself) is enough to obtain a clean class of graphs. Along the proof of the latter strengthening, we build on a result of Davies and produce, for every positive integer <em>η</em>, a complete description of unavoidable connected induced subgraphs of a connected graph <em>G</em> containing <em>η</em> vertices from a suitably large given set of vertices in <em>G</em>. This is of independent interest, and will be used in subsequent papers in this series.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"164 ","pages":"Pages 443-472"},"PeriodicalIF":1.4,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72364892","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":"Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree","authors":"Tara Abrishami ,&nbsp;Maria Chudnovsky ,&nbsp;Cemil Dibek ,&nbsp;Sepehr Hajebi ,&nbsp;Paweł Rzążewski ,&nbsp;Sophie Spirkl ,&nbsp;Kristina Vušković","doi":"10.1016/j.jctb.2023.10.005","DOIUrl":"10.1016/j.jctb.2023.10.005","url":null,"abstract":"<div><p><span><span>This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded </span>maximum degree, asserting that for all </span><em>k</em> and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of the <span><math><mo>(</mo><mi>k</mi><mo>×</mo><mi>k</mi><mo>)</mo></math></span>-wall or the line graph of a subdivision of the <span><math><mo>(</mo><mi>k</mi><mo>×</mo><mi>k</mi><mo>)</mo></math></span>-wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows.</p><ul><li><span>1.</span><span><p>For <span><math><mi>t</mi><mo>≥</mo><mn>2</mn></math></span>, a <em>t-theta</em> is a graph consisting of two nonadjacent vertices and three internally vertex-disjoint paths between them, each of length at least <em>t</em>. A <em>t-pyramid</em> is a graph consisting of a vertex <em>v</em>, a triangle <em>B</em> disjoint from <em>v</em> and three paths starting at <em>v</em> and vertex-disjoint otherwise, each joining <em>v</em> to a vertex of <em>B</em>, and each of length at least <em>t</em>. We prove that for all <span><math><mi>k</mi><mo>,</mo><mi>t</mi></math></span> and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a <em>t</em>-theta, or a <em>t</em>-pyramid, or the line graph of a subdivision of the <span><math><mo>(</mo><mi>k</mi><mo>×</mo><mi>k</mi><mo>)</mo></math></span>-wall as an induced subgraph. This affirmatively answers a question of Pilipczuk et al. asking whether every graph of bounded maximum degree and sufficiently large treewidth contains either a theta or a triangle as an induced subgraph (where a <em>theta</em> means a <em>t</em>-theta for some <span><math><mi>t</mi><mo>≥</mo><mn>2</mn></math></span>).</p></span></li><li><span>2.</span><span><p>A <em>subcubic subdivided caterpillar</em> is a tree of maximum degree at most three whose all vertices of degree three lie on a path. We prove that for every Δ and subcubic subdivided caterpillar <em>T</em>, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of <em>T</em> or the line graph of a subdivision of <em>T</em> as an induced subgraph.</p></span></li></ul></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"164 ","pages":"Pages 371-403"},"PeriodicalIF":1.4,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509768","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 recolouring version of Hadwiger's conjecture","authors":"Marthe Bonamy ,&nbsp;Marc Heinrich ,&nbsp;Clément Legrand-Duchesne ,&nbsp;Jonathan Narboni","doi":"10.1016/j.jctb.2023.10.001","DOIUrl":"10.1016/j.jctb.2023.10.001","url":null,"abstract":"<div><p>We prove that for any <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span>, for any large enough <em>t</em>, there is a graph that admits no <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>-minor but admits a <span><math><mo>(</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>ε</mi><mo>)</mo><mi>t</mi></math></span>-colouring that is “frozen” with respect to Kempe changes, i.e. any two colour classes induce a connected component. This disproves three conjectures of Las Vergnas and Meyniel from 1981.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"164 ","pages":"Pages 364-370"},"PeriodicalIF":1.4,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71513921","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":"Excluded minors for the Klein bottle I. Low connectivity case","authors":"Bojan Mohar ,&nbsp;Petr Škoda","doi":"10.1016/j.jctb.2023.10.002","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.10.002","url":null,"abstract":"<div><p>Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I we consider the structure of graphs with a 2-vertex-cut that are critical with respect to the Euler genus. A general theorem describing the building blocks is presented. These constituents, called hoppers and cascades, are classified for the case when Euler genus is small. As a consequence, the complete list of obstructions of connectivity 2 for embedding graphs into the Klein bottle is obtained. This is the first complete result about obstructions for embeddability of graphs in the Klein bottle, and the outcome is somewhat surprising in the sense that there are considerably fewer excluded minors than expected.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"164 ","pages":"Pages 299-320"},"PeriodicalIF":1.4,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50194444","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":"Quantum isomorphism of graphs from association schemes","authors":"Ada Chan ,&nbsp;William J. Martin","doi":"10.1016/j.jctb.2023.09.005","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.09.005","url":null,"abstract":"<div><p>We show that any two Hadamard graphs on the same number of vertices are quantum isomorphic. This follows from a more general recipe for showing quantum isomorphism of graphs arising from certain association schemes. The main result is built from three tools. A remarkable recent result <span>[20]</span> of Mančinska and Roberson shows that graphs <em>G</em> and <em>H</em> are quantum isomorphic if and only if, for any planar graph <em>F</em>, the number of graph homomorphisms from <em>F</em> to <em>G</em> is equal to the number of graph homomorphisms from <em>F</em> to <em>H</em>. A generalization of partition functions called “scaffolds” <span>[23]</span> affords some basic reduction rules such as series-parallel reduction and can be applied to counting homomorphisms. The final tool is the classical theorem of Epifanov showing that any plane graph can be reduced to a single vertex and no edges by extended series-parallel reductions and Delta-Wye transformations. This last sort of transformation is available to us in the case of exactly triply regular association schemes. The paper includes open problems and directions for future research.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"164 ","pages":"Pages 340-363"},"PeriodicalIF":1.4,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50194446","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":"Induced paths in graphs without anticomplete cycles","authors":"Tung Nguyen ,&nbsp;Alex Scott ,&nbsp;Paul Seymour","doi":"10.1016/j.jctb.2023.10.003","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.10.003","url":null,"abstract":"<div><p>Let us say a graph is <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span><em>-free</em>, where <span><math><mi>s</mi><mo>≥</mo><mn>1</mn></math></span> is an integer, if there do not exist <em>s</em> cycles of the graph that are pairwise vertex-disjoint and have no edges joining them. The structure of such graphs, even when <span><math><mi>s</mi><mo>=</mo><mn>2</mn></math></span>, is not well understood. For instance, until now we did not know how to test whether a graph is <span><math><msub><mrow><mi>O</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-free in polynomial time; and there was an open conjecture, due to Ngoc Khang Le, that <span><math><msub><mrow><mi>O</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-free graphs have only a polynomial number of induced paths.</p><p>In this paper we prove Le's conjecture; indeed, we will show that for all <span><math><mi>s</mi><mo>≥</mo><mn>1</mn></math></span>, there exists <span><math><mi>c</mi><mo>&gt;</mo><mn>0</mn></math></span> such that every <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>-free graph <em>G</em> has at most <span><math><mo>|</mo><mi>G</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>c</mi></mrow></msup></math></span> induced paths, where <span><math><mo>|</mo><mi>G</mi><mo>|</mo></math></span> is the number of vertices. This provides a poly-time algorithm to test if a graph is <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>-free, for all fixed <em>s</em>.</p><p>The proof has three parts. First, there is a short and beautiful proof, due to Le, that reduces the question to proving the same thing for graphs with no cycles of length four. Second, there is a recent result of Bonamy, Bonnet, Déprés, Esperet, Geniet, Hilaire, Thomassé and Wesolek, that in every <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>-free graph <em>G</em> with no cycle of length four, there is a set of vertices that intersects every cycle, with size logarithmic in <span><math><mo>|</mo><mi>G</mi><mo>|</mo></math></span>. And third, there is an argument that uses the result of Bonamy et al. to deduce the theorem. The last is the main content of this paper.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"164 ","pages":"Pages 321-339"},"PeriodicalIF":1.4,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50194445","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}