{"title":"Counting oriented trees in digraphs with large minimum semidegree","authors":"Felix Joos, Jonathan Schrodt","doi":"10.1016/j.jctb.2024.05.004","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.05.004","url":null,"abstract":"<div><p>Let <em>T</em> be an oriented tree on <em>n</em> vertices with maximum degree at most <span><math><msup><mrow><mi>e</mi></mrow><mrow><mi>o</mi><mo>(</mo><msqrt><mrow><mi>log</mi><mo></mo><mi>n</mi></mrow></msqrt><mo>)</mo></mrow></msup></math></span>. If <em>G</em> is a digraph on <em>n</em> vertices with minimum semidegree <span><math><msup><mrow><mi>δ</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mi>n</mi></math></span>, then <em>G</em> contains <em>T</em> as a spanning tree, as recently shown by Kathapurkar and Montgomery (in fact, they only require maximum degree <span><math><mi>o</mi><mo>(</mo><mi>n</mi><mo>/</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span>). This generalizes the corresponding result by Komlós, Sárközy and Szemerédi for graphs. We investigate the natural question how many copies of <em>T</em> the digraph <em>G</em> contains. Our main result states that every such <em>G</em> contains at least <span><math><mo>|</mo><mrow><mi>Aut</mi><mi>(</mi><mi>T</mi><mi>)</mi></mrow><mspace></mspace><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mi>n</mi><mo>!</mo></math></span> copies of <em>T</em>, which is optimal. This implies the analogous result in the undirected case.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"168 ","pages":"Pages 236-270"},"PeriodicalIF":1.4,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000431/pdfft?md5=7f84c56186e46b0ae787c373b4164785&pid=1-s2.0-S0095895624000431-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141244765","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 burning number conjecture holds asymptotically","authors":"Sergey Norin, Jérémie Turcotte","doi":"10.1016/j.jctb.2024.05.003","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.05.003","url":null,"abstract":"<div><p>The burning number <span><math><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a graph <em>G</em> is the smallest number of turns required to burn all vertices of a graph if at every turn a new fire is started and existing fires spread to all adjacent vertices. The Burning Number Conjecture of Bonato et al. (2016) postulates that <span><math><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mrow><mo>⌈</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>⌉</mo></mrow></math></span> for all connected graphs <em>G</em> on <em>n</em> vertices. We prove that this conjecture holds asymptotically, that is <span><math><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></math></span>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"168 ","pages":"Pages 208-235"},"PeriodicalIF":1.4,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141244764","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":"Directed cycles with zero weight in Zpk","authors":"Shoham Letzter , Natasha Morrison","doi":"10.1016/j.jctb.2024.05.002","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.05.002","url":null,"abstract":"<div><p>For a finite abelian group <em>A</em>, define <span><math><mi>f</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> to be the minimum integer such that for every complete digraph Γ on <em>f</em> vertices and every map <span><math><mi>w</mi><mo>:</mo><mi>E</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>→</mo><mi>A</mi></math></span>, there exists a directed cycle <em>C</em> in Γ such that <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>C</mi><mo>)</mo></mrow></msub><mi>w</mi><mo>(</mo><mi>e</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. The study of <span><math><mi>f</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> was initiated by Alon and Krivelevich (2021). In this article, we prove that <span><math><mi>f</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><mi>p</mi><mi>k</mi><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>k</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>, where <em>p</em> is prime, with an improved bound of <span><math><mi>O</mi><mo>(</mo><mi>k</mi><mi>log</mi><mo></mo><mi>k</mi><mo>)</mo></math></span> when <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span>. These bounds are tight up to a factor which is polylogarithmic in <em>k</em>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"168 ","pages":"Pages 192-207"},"PeriodicalIF":1.4,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000418/pdfft?md5=5e9d14a46eed8e2ee2946b39a3ab2037&pid=1-s2.0-S0095895624000418-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141244763","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":"Dirac-type theorems for long Berge cycles in hypergraphs","authors":"Alexandr Kostochka , Ruth Luo , Grace McCourt","doi":"10.1016/j.jctb.2024.05.001","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.05.001","url":null,"abstract":"<div><p>The famous Dirac's Theorem gives an exact bound on the minimum degree of an <em>n</em>-vertex graph guaranteeing the existence of a hamiltonian cycle. In the same paper, Dirac also observed that a graph with minimum degree at least <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> contains a cycle of length at least <span><math><mi>k</mi><mo>+</mo><mn>1</mn></math></span>. The purpose of this paper is twofold: we prove exact bounds of similar type for hamiltonian Berge cycles as well as for Berge cycles of length at least <em>k</em> in <em>r</em>-uniform, <em>n</em>-vertex hypergraphs for all combinations of <span><math><mi>k</mi><mo>,</mo><mi>r</mi></math></span> and <em>n</em> with <span><math><mn>3</mn><mo>≤</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span>. The bounds differ for different ranges of <em>r</em> compared to <em>n</em> and <em>k</em>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"168 ","pages":"Pages 159-191"},"PeriodicalIF":1.4,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141078547","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":"Dynamics of cycles in polyhedra I: The isolation lemma","authors":"Jan Kessler , Jens M. Schmidt","doi":"10.1016/j.jctb.2024.03.008","DOIUrl":"10.1016/j.jctb.2024.03.008","url":null,"abstract":"<div><div>A cycle <em>C</em> of a graph <em>G</em> is <em>isolating</em> if every component of <span><math><mi>G</mi><mo>−</mo><mi>V</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> consists of a single vertex. We show that isolating cycles in polyhedral graphs can be extended to larger ones: every isolating cycle <em>C</em> of length <span><math><mn>6</mn><mo>≤</mo><mo>|</mo><mi>E</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>|</mo><mo><</mo><mrow><mo>⌊</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>(</mo><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>+</mo><mn>4</mn><mo>)</mo><mo>⌋</mo></mrow></math></span> implies an isolating cycle <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of larger length that contains <span><math><mi>V</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span>. By “hopping” iteratively to such larger cycles, we obtain a powerful and very general inductive motor for proving long cycles and computing them (we will give an algorithm with quadratic running time). This is the first step towards the so far elusive quest of finding a universal induction that captures longest cycles of polyhedral graph classes.</div><div>Our motor provides also a method to prove linear lower bounds on the length of Tutte cycles, as <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> will be a Tutte cycle of <em>G</em> if <em>C</em> is. We prove in addition that <span><math><mo>|</mo><mi>E</mi><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo><mo>|</mo><mo>≤</mo><mo>|</mo><mi>E</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>|</mo><mo>+</mo><mn>3</mn></math></span> if <em>G</em><span> contains no face of size five, which gives a new tool for results about cycle spectra, and provides evidence that faces of size five may obstruct many different cycle lengths. As a sample application, we test our motor on the following so far unsettled conjecture about essentially 4-connected graphs.</span></div><div>A planar graph is <em>essentially</em> 4<em>-connected</em> if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Essentially 4-connected graphs have been thoroughly investigated throughout literature as the subject of Hamiltonicity studies. Jackson and Wormald proved that every essentially 4-connected planar graph <em>G</em> on <em>n</em> vertices contains a cycle of length at least <span><math><mfrac><mrow><mn>2</mn></mrow><mrow><mn>5</mn></mrow></mfrac><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo></math></span>, and this result has recently been improved multiple times, culminating in the lower bound <span><math><mfrac><mrow><mn>5</mn></mrow><mrow><mn>8</mn></mrow></mfrac><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo></math></span>. However, the currently best known upper bound is given by an infinite family of such graphs in which no graph <em>G</em> contains a cycle that is longer than <","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 329-364"},"PeriodicalIF":1.2,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143894862","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}
Linda Cook , Jake Horsfield , Myriam Preissmann , Cléophée Robin , Paul Seymour , Ni Luh Dewi Sintiari , Nicolas Trotignon , Kristina Vušković
{"title":"Graphs with all holes the same length","authors":"Linda Cook , Jake Horsfield , Myriam Preissmann , Cléophée Robin , Paul Seymour , Ni Luh Dewi Sintiari , Nicolas Trotignon , Kristina Vušković","doi":"10.1016/j.jctb.2024.04.006","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.04.006","url":null,"abstract":"<div><p>A graph is <em>ℓ-holed</em> if all its induced cycles of length at least four have length exactly <em>ℓ</em>. We give a complete description of the <em>ℓ</em>-holed graphs for each <span><math><mi>ℓ</mi><mo>≥</mo><mn>7</mn></math></span>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"168 ","pages":"Pages 96-158"},"PeriodicalIF":1.4,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140924613","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}
Gabriel Coutinho, Emanuel Juliano, Thomás Jung Spier
{"title":"No perfect state transfer in trees with more than 3 vertices","authors":"Gabriel Coutinho, Emanuel Juliano, Thomás Jung Spier","doi":"10.1016/j.jctb.2024.04.004","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.04.004","url":null,"abstract":"<div><p>We prove that the only trees that admit perfect state transfer according to the adjacency matrix model are <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>. This answers a question first asked by Godsil in 2012 and proves a conjecture by Coutinho and Liu from 2015.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"168 ","pages":"Pages 68-85"},"PeriodicalIF":1.4,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140901144","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}
António Girão , Kevin Hendrey , Freddie Illingworth , Florian Lehner , Lukas Michel , Michael Savery , Raphael Steiner
{"title":"Chromatic number is not tournament-local","authors":"António Girão , Kevin Hendrey , Freddie Illingworth , Florian Lehner , Lukas Michel , Michael Savery , Raphael Steiner","doi":"10.1016/j.jctb.2024.04.005","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.04.005","url":null,"abstract":"<div><p>Scott and Seymour conjectured the existence of a function <span><math><mi>f</mi><mo>:</mo><mi>N</mi><mo>→</mo><mi>N</mi></math></span> such that, for every graph <em>G</em> and tournament <em>T</em> on the same vertex set, <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>⩾</mo><mi>f</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> implies that <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>[</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>T</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mi>v</mi><mo>)</mo><mo>]</mo><mo>)</mo><mo>⩾</mo><mi>k</mi></math></span> for some vertex <em>v</em>. In this note we disprove this conjecture even if <em>v</em> is replaced by a vertex set of size <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo></mo><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>)</mo></math></span>. As a consequence, we answer in the negative a question of Harutyunyan, Le, Thomassé, and Wu concerning the corresponding statement where the graph <em>G</em> is replaced by another tournament, and disprove a related conjecture of Nguyen, Scott, and Seymour. We also show that the setting where chromatic number is replaced by degeneracy exhibits a quite different behaviour.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"168 ","pages":"Pages 86-95"},"PeriodicalIF":1.4,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000315/pdfft?md5=96fc5d216231691bd8b06b1f5ac0f4bd&pid=1-s2.0-S0095895624000315-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140900886","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":"Embedding loose spanning trees in 3-uniform hypergraphs","authors":"Yanitsa Pehova , Kalina Petrova","doi":"10.1016/j.jctb.2024.04.003","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.04.003","url":null,"abstract":"<div><p>In 1995, Komlós, Sárközy and Szemerédi showed that every large <em>n</em>-vertex graph with minimum degree at least <span><math><mo>(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>γ</mi><mo>)</mo><mi>n</mi></math></span> contains all spanning trees of bounded degree. We consider a generalization of this result to loose spanning hypertrees in 3-graphs, that is, linear hypergraphs obtained by successively appending edges sharing a single vertex with a previous edge. We show that for all <em>γ</em> and Δ, and <em>n</em> large, every <em>n</em>-vertex 3-uniform hypergraph of minimum vertex degree <span><math><mo>(</mo><mn>5</mn><mo>/</mo><mn>9</mn><mo>+</mo><mi>γ</mi><mo>)</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span> contains every loose spanning tree <em>T</em> with maximum vertex degree Δ. This bound is asymptotically tight, since some loose trees contain perfect matchings.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"168 ","pages":"Pages 47-67"},"PeriodicalIF":1.4,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000303/pdfft?md5=4e333586884c0a88ecc3b2284d18ce92&pid=1-s2.0-S0095895624000303-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140880459","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 inducibility of oriented stars","authors":"Ping Hu , Jie Ma , Sergey Norin , Hehui Wu","doi":"10.1016/j.jctb.2024.04.001","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.04.001","url":null,"abstract":"<div><p>We consider the problem of maximizing the number of induced copies of an oriented star <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub></math></span> in digraphs of given size, where the center of the star has out-degree <em>k</em> and in-degree <em>ℓ</em>. The case <span><math><mi>k</mi><mi>ℓ</mi><mo>=</mo><mn>0</mn></math></span> was solved by Huang in <span>[11]</span>. Here, we asymptotically solve it for all other oriented stars with at least seven vertices.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"168 ","pages":"Pages 11-46"},"PeriodicalIF":1.4,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140843070","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}
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