{"title":"Turán problems for mixed graphs","authors":"Nitya Mani , Edward Yu","doi":"10.1016/j.jctb.2024.02.004","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.02.004","url":null,"abstract":"<div><p>We investigate natural Turán problems for mixed graphs, generalizations of graphs where edges can be either directed or undirected. We study a natural <em>Turán density coefficient</em> that measures how large a fraction of directed edges an <em>F</em>-free mixed graph can have; we establish an analogue of the Erdős-Stone-Simonovits theorem and give a variational characterization of the Turán density coefficient of any mixed graph (along with an associated extremal <em>F</em>-free family).</p><p>This characterization enables us to highlight an important divergence between classical extremal numbers and the Turán density coefficient. We show that Turán density coefficients can be irrational, but are always algebraic; for every positive integer <em>k</em>, we construct a family of mixed graphs whose Turán density coefficient has algebraic degree <em>k</em>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"167 ","pages":"Pages 119-163"},"PeriodicalIF":1.4,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140145159","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":"Finite matchability under the matroidal Hall's condition","authors":"Attila Joó","doi":"10.1016/j.jctb.2024.02.006","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.02.006","url":null,"abstract":"<div><p>Aharoni and Ziv conjectured that if <em>M</em> and <em>N</em> are finitary matroids on <em>E</em>, then a certain “Hall-like” condition is sufficient to guarantee the existence of an <em>M</em>-independent spanning set of <em>N</em>. We show that their condition ensures that every finite subset of <em>E</em> is <em>N</em>-spanned by an <em>M</em>-independent set.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"167 ","pages":"Pages 104-118"},"PeriodicalIF":1.4,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000121/pdfft?md5=545e85909eb190e88b95a350a595c764&pid=1-s2.0-S0095895624000121-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140121830","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":"Cycle decompositions in k-uniform hypergraphs","authors":"Allan Lo , Simón Piga , Nicolás Sanhueza-Matamala","doi":"10.1016/j.jctb.2024.02.003","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.02.003","url":null,"abstract":"<div><p>We show that <em>k</em>-uniform hypergraphs on <em>n</em> vertices whose codegree is at least <span><math><mo>(</mo><mn>2</mn><mo>/</mo><mn>3</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mi>n</mi></math></span> can be decomposed into tight cycles, subject to the trivial divisibility conditions. As a corollary, we show those graphs contain tight Euler tours as well. In passing, we also investigate decompositions into tight paths.</p><p>In addition, we also prove an alternative condition for building absorbers for edge-decompositions of arbitrary <em>k</em>-uniform hypergraphs, which should be of independent interest.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"167 ","pages":"Pages 55-103"},"PeriodicalIF":1.4,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000091/pdfft?md5=4e1edb0999ca3378e8c423d7ea50f42a&pid=1-s2.0-S0095895624000091-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140041359","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 theorems for even cycles in random hypergraph","authors":"Jiaxi Nie","doi":"10.1016/j.jctb.2024.02.002","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.02.002","url":null,"abstract":"<div><p>Let <span><math><mi>F</mi></math></span> be a family of <em>r</em>-uniform hypergraphs. The random Turán number <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>,</mo><mi>F</mi><mo>)</mo></math></span> is the maximum number of edges in an <span><math><mi>F</mi></math></span>-free subgraph of <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>, where <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span> is the Erdős-Rényi random <em>r</em>-graph with parameter <em>p</em>. Let <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>ℓ</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span> denote the <em>r</em>-uniform linear cycle of length <em>ℓ</em>. For <span><math><mi>p</mi><mo>≥</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>r</mi><mo>+</mo><mn>2</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>, Mubayi and Yepremyan showed that <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mi>max</mi><mo></mo><mo>{</mo><msup><mrow><mi>p</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup><mo>×</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>,</mo><mi>p</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>}</mo></math></span>. This upper bound is not tight when <span><math><mi>p</mi><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>r</mi><mo>+</mo><mn>2</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>. In this paper, we close the gap for <span><math><mi>r</mi><mo>≥</mo><mn>4</mn></math></span>. More precisely, we show that <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>)</mo><mo>=</mo><mi>Θ</mi><mo>(</mo><mi>p</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> when <span><math><mi>p</mi><mo>≥</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>r</mi><mo>+</mo><m","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"167 ","pages":"Pages 23-54"},"PeriodicalIF":1.4,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139999287","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}
Zequn Lv , Ervin Győri , Zhen He , Nika Salia , Casey Tompkins , Xiutao Zhu
{"title":"The maximum number of copies of an even cycle in a planar graph","authors":"Zequn Lv , Ervin Győri , Zhen He , Nika Salia , Casey Tompkins , Xiutao Zhu","doi":"10.1016/j.jctb.2024.01.005","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.01.005","url":null,"abstract":"<div><p>We resolve a conjecture of Cox and Martin by determining asymptotically for every <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> the maximum number of copies of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub></math></span> in an <em>n</em>-vertex planar graph.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"167 ","pages":"Pages 15-22"},"PeriodicalIF":1.4,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139936460","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":"How connectivity affects the extremal number of trees","authors":"Suyun Jiang , Hong Liu , Nika Salia","doi":"10.1016/j.jctb.2024.02.001","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.02.001","url":null,"abstract":"<div><p>The Erdős-Sós conjecture states that the maximum number of edges in an <em>n</em>-vertex graph without a given <em>k</em>-vertex tree is at most <span><math><mfrac><mrow><mi>n</mi><mo>(</mo><mi>k</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. Despite significant interest, the conjecture remains unsolved. Recently, Caro, Patkós, and Tuza considered this problem for host graphs that are connected. Settling a problem posed by them, for a <em>k</em>-vertex tree <em>T</em>, we construct <em>n</em>-vertex connected graphs that are <em>T</em>-free with at least <span><math><mo>(</mo><mn>1</mn><mo>/</mo><mn>4</mn><mo>−</mo><msub><mrow><mi>o</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mi>n</mi><mi>k</mi></math></span> edges, showing that the additional connectivity condition can reduce the maximum size by at most a factor of 2. Furthermore, we show that this is optimal: there is a family of <em>k</em>-vertex brooms <em>T</em> such that the maximum size of an <em>n</em>-vertex connected <em>T</em>-free graph is at most <span><math><mo>(</mo><mn>1</mn><mo>/</mo><mn>4</mn><mo>+</mo><msub><mrow><mi>o</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mi>n</mi><mi>k</mi></math></span>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"167 ","pages":"Pages 1-14"},"PeriodicalIF":1.4,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139907785","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":"A solution to the 1-2-3 conjecture","authors":"Ralph Keusch","doi":"10.1016/j.jctb.2024.01.002","DOIUrl":"10.1016/j.jctb.2024.01.002","url":null,"abstract":"<div><p>We show that for every graph without isolated edge, the edges can be assigned weights from <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo></math></span> so that no two neighbors receive the same sum of incident edge weights. This solves a conjecture of Karoński, Łuczak, and Thomason from 2004.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"166 ","pages":"Pages 183-202"},"PeriodicalIF":1.4,"publicationDate":"2024-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139566010","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":"The minimum degree removal lemma thresholds","authors":"Lior Gishboliner, Zhihan Jin, Benny Sudakov","doi":"10.1016/j.jctb.2024.01.003","DOIUrl":"10.1016/j.jctb.2024.01.003","url":null,"abstract":"<div><p>The graph removal lemma is a fundamental result in extremal graph theory which says that for every fixed graph <em>H</em> and <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span>, if an <em>n</em>-vertex graph <em>G</em> contains <span><math><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> edge-disjoint copies of <em>H</em> then <em>G</em> contains <span><math><mi>δ</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>v</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></msup></math></span> copies of <em>H</em> for some <span><math><mi>δ</mi><mo>=</mo><mi>δ</mi><mo>(</mo><mi>ε</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>></mo><mn>0</mn></math></span>. The current proofs of the removal lemma give only very weak bounds on <span><math><mi>δ</mi><mo>(</mo><mi>ε</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>, and it is also known that <span><math><mi>δ</mi><mo>(</mo><mi>ε</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is not polynomial in <em>ε</em> unless <em>H</em> is bipartite. Recently, Fox and Wigderson initiated the study of minimum degree conditions guaranteeing that <span><math><mi>δ</mi><mo>(</mo><mi>ε</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> depends polynomially or linearly on <em>ε</em>. In this paper we answer several questions of Fox and Wigderson on this topic.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"166 ","pages":"Pages 203-221"},"PeriodicalIF":1.4,"publicationDate":"2024-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000042/pdfft?md5=933ffe8670f6d93ed7560c5af633e7e3&pid=1-s2.0-S0095895624000042-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139567936","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}
Yan Cao , Guantao Chen , Guangming Jing , Songling Shan
{"title":"The core conjecture of Hilton and Zhao","authors":"Yan Cao , Guantao Chen , Guangming Jing , Songling Shan","doi":"10.1016/j.jctb.2024.01.004","DOIUrl":"10.1016/j.jctb.2024.01.004","url":null,"abstract":"<div><p>A simple graph <em>G</em><span> with maximum degree Δ is </span><em>overfull</em> if <span><math><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>></mo><mi>Δ</mi><mo>⌊</mo><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>/</mo><mn>2</mn><mo>⌋</mo></math></span>. The <em>core</em> of <em>G</em>, denoted <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>Δ</mi></mrow></msub></math></span>, is the subgraph of <em>G</em> induced by its vertices of degree Δ. Clearly, the chromatic index of <em>G</em> equals <span><math><mi>Δ</mi><mo>+</mo><mn>1</mn></math></span> if <em>G</em> is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if <em>G</em><span> is a simple connected graph with </span><span><math><mi>Δ</mi><mo>≥</mo><mn>3</mn></math></span> and <span><math><mi>Δ</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>Δ</mi></mrow></msub><mo>)</mo><mo>≤</mo><mn>2</mn></math></span>, then <span><math><msup><mrow><mi>χ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Δ</mi><mo>+</mo><mn>1</mn></math></span> implies that <em>G</em> is overfull or <span><math><mi>G</mi><mo>=</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, where <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is obtained from the Petersen graph by deleting a vertex. Cariolaro and Cariolaro settled the base case <span><math><mi>Δ</mi><mo>=</mo><mn>3</mn></math></span> in 2003, and Cranston and Rabern proved the next case, <span><math><mi>Δ</mi><mo>=</mo><mn>4</mn></math></span>, in 2019. In this paper, we give a proof of this conjecture for all <span><math><mi>Δ</mi><mo>≥</mo><mn>4</mn></math></span>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"166 ","pages":"Pages 154-182"},"PeriodicalIF":1.4,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139551178","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 orders of automorphisms of vertex-transitive graphs","authors":"Primož Potočnik , Micael Toledo , Gabriel Verret","doi":"10.1016/j.jctb.2024.01.001","DOIUrl":"10.1016/j.jctb.2024.01.001","url":null,"abstract":"<div><p>In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with <em>n</em> vertices and of valence <em>d</em>, <span><math><mi>d</mi><mo>≤</mo><mn>4</mn></math></span>, is at most <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msub><mi>n</mi></math></span> where <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span> and <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>=</mo><mn>9</mn></math></span>. Whether such a constant <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> exists for valencies larger than 4 remains an unanswered question. Further, we prove that every automorphism <em>g</em> of a finite connected 3-valent vertex-transitive graph Γ, <span><math><mi>Γ</mi><mo>≇</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>3</mn></mrow></msub></math></span>, has a regular orbit, that is, an orbit of <span><math><mo>〈</mo><mi>g</mi><mo>〉</mo></math></span> of length equal to the order of <em>g</em>. Moreover, we prove that in this case either Γ belongs to a well understood family of exceptional graphs or at least 5/12 of the vertices of Γ belong to a regular orbit of <em>g</em>. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms <em>C</em> of a connected 3-valent vertex-transitive graph Γ in terms of the number of vertices of Γ and the length of a longest orbit of <em>C</em>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"166 ","pages":"Pages 123-153"},"PeriodicalIF":1.4,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000029/pdfft?md5=5bdead4227eef1a873304cc296bf7df1&pid=1-s2.0-S0095895624000029-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139522954","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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