Journal of Combinatorial Theory Series B最新文献

{"title":"A half-integral Erdős-Pósa theorem for directed odd cycles","authors":"Ken-ichi Kawarabayashi ,&nbsp;Stephan Kreutzer ,&nbsp;O-joung Kwon ,&nbsp;Qiqin Xie","doi":"10.1016/j.jctb.2024.12.008","DOIUrl":"10.1016/j.jctb.2024.12.008","url":null,"abstract":"<div><div>We prove that there exists a function <span><math><mi>f</mi><mo>:</mo><mi>N</mi><mo>→</mo><mi>R</mi></math></span> such that every directed graph <em>G</em> contains either <em>k</em> directed odd cycles where every vertex of <em>G</em> is contained in at most two of them, or a set of at most <span><math><mi>f</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> vertices meeting all directed odd cycles. We give a polynomial-time algorithm for fixed <em>k</em> which outputs one of the two outcomes. This extends the half-integral Erdős-Pósa theorem for undirected odd cycles by Reed [Combinatorica 1999] to directed graphs.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 115-145"},"PeriodicalIF":1.2,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092867","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 the automorphism group of a distance-regular graph","authors":"László Pyber ,&nbsp;Saveliy V. Skresanov","doi":"10.1016/j.jctb.2024.12.005","DOIUrl":"10.1016/j.jctb.2024.12.005","url":null,"abstract":"<div><div>The motion of a graph is the minimal degree of its full automorphism group. Babai conjectured that the motion of a primitive distance-regular graph on <em>n</em> vertices of diameter greater than two is at least <span><math><mi>n</mi><mo>/</mo><mi>C</mi></math></span> for some universal constant <span><math><mi>C</mi><mo>&gt;</mo><mn>0</mn></math></span>, unless the graph is a Johnson or Hamming graph. We prove that the motion of a distance-regular graph of diameter <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> on <em>n</em> vertices is at least <span><math><mi>C</mi><mi>n</mi><mo>/</mo><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>6</mn></mrow></msup></math></span> for some universal constant <span><math><mi>C</mi><mo>&gt;</mo><mn>0</mn></math></span>, unless it is a Johnson, Hamming or crown graph. To show this, we improve an earlier result by Kivva who gave a lower bound on motion of the form <span><math><mi>n</mi><mo>/</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span>, where <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> depends exponentially on <em>d</em>. As a corollary we derive a quasipolynomial upper bound for the size of the automorphism group of a primitive distance-regular graph acting edge-transitively on the graph and on its distance-2 graph. The proofs use elementary combinatorial arguments and do not depend on the classification of finite simple groups.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 94-114"},"PeriodicalIF":1.2,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092866","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":"Aharoni's rainbow cycle conjecture holds up to an additive constant","authors":"Patrick Hompe,&nbsp;Tony Huynh","doi":"10.1016/j.jctb.2024.12.004","DOIUrl":"10.1016/j.jctb.2024.12.004","url":null,"abstract":"<div><div>In 2017, Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture: if <em>G</em> is a simple <em>n</em>-vertex edge-colored graph with <em>n</em> color classes of size at least <em>r</em>, then <em>G</em> contains a rainbow cycle of length at most <span><math><mo>⌈</mo><mi>n</mi><mo>/</mo><mi>r</mi><mo>⌉</mo></math></span>.</div><div>In this paper, we prove that, for fixed <em>r</em>, Aharoni's conjecture holds up to an additive constant. Specifically, we show that for each fixed <span><math><mi>r</mi><mo>⩾</mo><mn>1</mn></math></span>, there exists a constant <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>∈</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>r</mi></mrow><mrow><mn>5</mn></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⁡</mo><mi>r</mi><mo>)</mo></math></span> such that if <em>G</em> is a simple <em>n</em>-vertex edge-colored graph with <em>n</em> color classes of size at least <em>r</em>, then <em>G</em> contains a rainbow cycle of length at most <span><math><mi>n</mi><mo>/</mo><mi>r</mi><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 80-93"},"PeriodicalIF":1.2,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092865","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":"Slow graph bootstrap percolation II: Accelerating properties","authors":"David Fabian ,&nbsp;Patrick Morris ,&nbsp;Tibor Szabó","doi":"10.1016/j.jctb.2024.12.006","DOIUrl":"10.1016/j.jctb.2024.12.006","url":null,"abstract":"<div><div>For a graph <em>H</em> and an <em>n</em>-vertex graph <em>G</em>, the <em>H</em>-bootstrap process on <em>G</em> is the process which starts with <em>G</em> and, at every time step, adds any missing edges on the vertices of <em>G</em> that complete a copy of <em>H</em>. This process eventually stabilises and we are interested in the extremal question raised by Bollobás of determining the maximum <em>running time</em> (number of time steps before stabilising) of this process over all possible choices of <em>n</em>-vertex graph <em>G</em>. In this paper, we initiate a systematic study of the asymptotics of this parameter, denoted <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, and its dependence on properties of the graph <em>H</em>. Our focus is on <em>H</em> which define relatively fast bootstrap processes, that is, with <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> being at most linear in <em>n</em>. We study the graph class of trees, showing that one can bound <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> by a quadratic function in <span><math><mi>v</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> for all trees <em>T</em> and all <em>n</em>. We then go on to explore the relationship between the running time of the <em>H</em>-process and the minimum vertex degree and connectivity of <em>H</em>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 44-79"},"PeriodicalIF":1.2,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143127972","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":"Unexpected automorphisms in direct product graphs","authors":"Yunsong Gan ,&nbsp;Weijun Liu ,&nbsp;Binzhou Xia","doi":"10.1016/j.jctb.2024.12.003","DOIUrl":"10.1016/j.jctb.2024.12.003","url":null,"abstract":"<div><div>A pair of graphs <span><math><mo>(</mo><mi>Γ</mi><mo>,</mo><mi>Σ</mi><mo>)</mo></math></span> is called unstable if their direct product <span><math><mi>Γ</mi><mo>×</mo><mi>Σ</mi></math></span> has automorphisms that do not come from <span><math><mtext>Aut</mtext><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>×</mo><mtext>Aut</mtext><mo>(</mo><mi>Σ</mi><mo>)</mo></math></span>, and such automorphisms are said to be unexpected. In the special case when <span><math><mi>Σ</mi><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, the stability of <span><math><mo>(</mo><mi>Γ</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> is well studied in the literature, where the so-called two-fold automorphisms of the graph Γ have played an important role. As a generalization of two-fold automorphisms, the concept of non-diagonal automorphisms was recently introduced to study the stability of general graph pairs. In this paper, we obtain, for a large family of graph pairs, a necessary and sufficient condition to be unstable in terms of the existence of non-diagonal automorphisms. As a byproduct, we determine the stability of graph pairs involving complete graphs or odd cycles, respectively. The former result in fact solves a problem proposed by Dobson, Miklavič and Šparl for undirected graphs, as well as confirms a recent conjecture of Qin, Xia and Zhou.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"171 ","pages":"Pages 140-164"},"PeriodicalIF":1.2,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142929318","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":"Intersecting families with covering number three","authors":"Peter Frankl ,&nbsp;Jian Wang","doi":"10.1016/j.jctb.2024.12.001","DOIUrl":"10.1016/j.jctb.2024.12.001","url":null,"abstract":"<div><div>We consider <em>k</em>-graphs on <em>n</em> vertices, that is, <span><math><mi>F</mi><mo>⊂</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. A <em>k</em>-graph <span><math><mi>F</mi></math></span> is called intersecting if <span><math><mi>F</mi><mo>∩</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≠</mo><mo>∅</mo></math></span> for all <span><math><mi>F</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>F</mi></math></span>. In the present paper we prove that for <span><math><mi>k</mi><mo>≥</mo><mn>7</mn></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>k</mi></math></span>, any intersecting <em>k</em>-graph <span><math><mi>F</mi></math></span> with covering number at least three, satisfies <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>2</mn><mi>k</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mn>3</mn></math></span>, the best possible upper bound which was proved in <span><span>[4]</span></span> subject to exponential constraints <span><math><mi>n</mi><mo>&gt;</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"171 ","pages":"Pages 96-139"},"PeriodicalIF":1.2,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142929317","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":"Invariants of Tutte partitions and a q-analogue","authors":"Eimear Byrne,&nbsp;Andrew Fulcher","doi":"10.1016/j.jctb.2024.12.002","DOIUrl":"10.1016/j.jctb.2024.12.002","url":null,"abstract":"<div><div>We describe a construction of the Tutte polynomial for both matroids and <em>q</em>-matroids based on an appropriate partition of the underlying support lattice into intervals that correspond to prime-free minors, which we call a Tutte partition. We show that such partitions in the matroid case include the class of partitions arising in Crapo's definition of the Tutte polynomial, while not representing a direct <em>q</em>-analogue of such partitions. We propose axioms of a <em>q</em>-Tutte-Grothendieck invariant and show that this yields a <em>q</em>-analogue of a Tutte-Grothendieck invariant. We establish the connection between the rank generating polynomial and the Tutte polynomial, showing that one can be obtained from the other by convolution.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 1-43"},"PeriodicalIF":1.2,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092945","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":"Orientably-regular embeddings of complete multigraphs","authors":"Štefan Gyürki,&nbsp;Soňa Pavlíková,&nbsp;Jozef Širáň","doi":"10.1016/j.jctb.2024.11.004","DOIUrl":"10.1016/j.jctb.2024.11.004","url":null,"abstract":"<div><div>An embedding of a graph on an orientable surface is <em>orientably-regular</em> (or <em>rotary</em>, in an equivalent terminology) if the group of orientation-preserving automorphisms of the embedding is transitive (and hence regular) on incident vertex-edge pairs of the graph. A classification of orientably-regular embeddings of complete graphs was obtained by L.D. James and G.A. Jones (1985) <span><span>[10]</span></span>, pointing out interesting connections to finite fields and Frobenius groups. By a combination of graph-theoretic methods and tools from combinatorial group theory we extend results of James and Jones to classification of orientably-regular embeddings of complete multigraphs with arbitrary edge-multiplicity.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"171 ","pages":"Pages 71-95"},"PeriodicalIF":1.2,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142929316","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 conjecture of Tokushige for cross-t-intersecting families","authors":"Huajun Zhang ,&nbsp;Biao Wu","doi":"10.1016/j.jctb.2024.11.005","DOIUrl":"10.1016/j.jctb.2024.11.005","url":null,"abstract":"<div><div>Two families of sets <span><math><mi>A</mi></math></span> and <span><math><mi>B</mi></math></span> are called cross-<em>t</em>-intersecting if <span><math><mo>|</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>|</mo><mo>≥</mo><mi>t</mi></math></span> for all <span><math><mi>A</mi><mo>∈</mo><mi>A</mi></math></span>, <span><math><mi>B</mi><mo>∈</mo><mi>B</mi></math></span>. An active problem in extremal set theory is to determine the maximum product of sizes of cross-<em>t</em>-intersecting families. This incorporates the classical Erdős–Ko–Rado (EKR) problem. In the present paper, we prove that if <span><math><mi>A</mi><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> and <span><math><mi>B</mi><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> are cross-<em>t</em>-intersecting with <span><math><mi>k</mi><mo>≥</mo><mi>t</mi><mo>≥</mo><mn>3</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>k</mi><mo>−</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>, then <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>|</mo><mi>B</mi><mo>|</mo><mo>≤</mo><msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>t</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Moreover, equality holds if and only if <span><math><mi>A</mi><mo>=</mo><mi>B</mi></math></span> is a maximum <em>t</em>-intersecting subfamily of <span><math><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></math></span>. This confirms a conjecture of Tokushige for <span><math><mi>t</mi><mo>≥</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"171 ","pages":"Pages 49-70"},"PeriodicalIF":1.2,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142929320","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":"Linear three-uniform hypergraphs with no Berge path of given length","authors":"Ervin Győri ,&nbsp;Nika Salia","doi":"10.1016/j.jctb.2024.11.003","DOIUrl":"10.1016/j.jctb.2024.11.003","url":null,"abstract":"<div><div>Extensions of Erdős-Gallai Theorem for general hypergraphs are well studied. In this work, we prove the extension of Erdős-Gallai Theorem for linear hypergraphs. In particular, we show that the number of hyperedges in an <em>n</em>-vertex 3-uniform linear hypergraph, without a Berge path of length <em>k</em> as a subgraph is at most <span><math><mfrac><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>6</mn></mrow></mfrac><mi>n</mi></math></span> for <span><math><mi>k</mi><mo>≥</mo><mn>4</mn></math></span>. The bound is sharp for infinitely many <em>k</em> and <em>n</em>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"171 ","pages":"Pages 36-48"},"PeriodicalIF":1.2,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142929321","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}