{"title":"Turán graphs with bounded matching number","authors":"Noga Alon , 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":"165 ","pages":"Pages 223-229"},"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 , Alex Scott , 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":"165 ","pages":"Pages 211-222"},"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 , 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":"165 ","pages":"Pages 197-210"},"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 , Yanjia Li , 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":"165 ","pages":"Pages 142-163"},"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 , Piotr Micek , 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":"165 ","pages":"Pages 164-196"},"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 , Nicolás Sanhueza-Matamala , 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":"165 ","pages":"Pages 97-141"},"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}
Marthe Bonamy , Michelle Delcourt , Richard Lang , Luke Postle
{"title":"Edge-colouring graphs with local list sizes","authors":"Marthe Bonamy , Michelle Delcourt , Richard Lang , 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":"165 ","pages":"Pages 68-96"},"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}
{"title":"Generalized cut trees for edge-connectivity","authors":"On-Hei Solomon Lo , Jens M. Schmidt","doi":"10.1016/j.jctb.2023.11.003","DOIUrl":"10.1016/j.jctb.2023.11.003","url":null,"abstract":"<div><p>We present three cut trees of graphs, each of them giving insights into the edge-connectivity structure. All three cut trees have in common that they are defined with respect to a given binary symmetric relation <em>R</em> on the vertex set of the graph, which generalizes Gomory-Hu trees. Applying these cut trees, we prove the following:</p><ul><li><span>•</span><span><p>A pair of vertices <span><math><mo>{</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo>}</mo></math></span> of a graph <em>G</em> is <em>pendant</em> if <span><math><mi>λ</mi><mo>(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>=</mo><mi>min</mi><mo></mo><mo>{</mo><mi>d</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>,</mo><mi>d</mi><mo>(</mo><mi>w</mi><mo>)</mo><mo>}</mo></math></span>. Mader showed in 1974 that every simple graph with minimum degree <em>δ</em> contains at least <span><math><mi>δ</mi><mo>(</mo><mi>δ</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span> pendant pairs. We improve this lower bound to <span><math><mi>δ</mi><mi>n</mi><mo>/</mo><mn>24</mn></math></span> for every simple graph <em>G</em> on <em>n</em> vertices with <span><math><mi>δ</mi><mo>≥</mo><mn>5</mn></math></span> or <span><math><mi>λ</mi><mo>≥</mo><mn>4</mn></math></span> or vertex connectivity <span><math><mi>κ</mi><mo>≥</mo><mn>3</mn></math></span>, and show that this is optimal up to a constant factor with regard to every parameter.</p></span></li><li><span>•</span><span><p>Every simple graph <em>G</em> satisfying <span><math><mi>δ</mi><mo>></mo><mn>0</mn></math></span> has <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>/</mo><mi>δ</mi><mo>)</mo></math></span> <em>δ</em>-edge-connected components. Moreover, every simple graph <em>G</em> that satisfies <span><math><mn>0</mn><mo><</mo><mi>λ</mi><mo><</mo><mi>δ</mi></math></span> has <span><math><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>n</mi><mo>/</mo><mi>δ</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> cuts of size less than <span><math><mi>min</mi><mo></mo><mo>{</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>λ</mi><mo>,</mo><mi>δ</mi><mo>}</mo></math></span>, and <span><math><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>n</mi><mo>/</mo><mi>δ</mi><mo>)</mo></mrow><mrow><mo>⌊</mo><mn>2</mn><mi>α</mi><mo>⌋</mo></mrow></msup><mo>)</mo></math></span> cuts of size at most <span><math><mi>min</mi><mo></mo><mo>{</mo><mi>α</mi><mo>⋅</mo><mi>λ</mi><mo>,</mo><mi>δ</mi><mo>−</mo><mn>1</mn><mo>}</mo></math></span> for any given real number <span><math><mi>α</mi><mo>≥</mo><mn>1</mn></math></span>.</p></span></li><li><span>•</span><span><p>A cut is <em>trivial</em> if it or its complement in <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is a singleton. We provide an alternative proof of the following recent result of Lo et al.: Given a simple graph <em>G</em> on <em>n</em> vertices that satisfies <span><math><mi>δ</mi><mo>></mo><mn>0</mn></math></span>, we can compu","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"165 ","pages":"Pages 47-67"},"PeriodicalIF":1.4,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138289378","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":"Three-coloring triangle-free graphs on surfaces VI. 3-colorability of quadrangulations","authors":"Zdeněk Dvořák , Daniel Král' , Robin Thomas","doi":"10.1016/j.jctb.2023.11.001","DOIUrl":"10.1016/j.jctb.2023.11.001","url":null,"abstract":"<div><p>We give a linear-time algorithm to decide 3-colorability (and find a 3-coloring, if it exists) of quadrangulations of a fixed surface. The algorithm also allows to prescribe the coloring for a bounded number of vertices.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"164 ","pages":"Pages 517-548"},"PeriodicalIF":1.4,"publicationDate":"2023-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138293164","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":"Maximal matroids in weak order posets","authors":"Bill Jackson , Shin-ichi Tanigawa","doi":"10.1016/j.jctb.2023.10.012","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.10.012","url":null,"abstract":"<div><p>Let <span><math><mi>X</mi></math></span> be a family of subsets of a finite set <em>E</em>. A matroid on <em>E</em> is called an <span><math><mi>X</mi></math></span>-matroid if each set in <span><math><mi>X</mi></math></span> is a circuit. We develop techniques for determining when there exists a unique maximal <span><math><mi>X</mi></math></span>-matroid in the weak order poset of all <span><math><mi>X</mi></math></span>-matroids on <em>E</em> and formulate a conjecture which would characterise the rank function of this unique maximal matroid when it exists. The conjecture suggests a new type of matroid rank function which extends the concept of weakly saturated sequences from extremal graph theory. We verify the conjecture for various families <span><math><mi>X</mi></math></span> and show that, if true, the conjecture could have important applications in such areas as combinatorial rigidity and low rank matrix completion.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"165 ","pages":"Pages 20-46"},"PeriodicalIF":1.4,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895623000928/pdfft?md5=d5b2f8e0d2e06aed011e9b1335503fd6&pid=1-s2.0-S0095895623000928-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138396187","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}
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