Journal of Combinatorial Theory Series B最新文献

{"title":"Kővári-Sós-Turán theorem for hereditary families","authors":"Zach Hunter ,&nbsp;Aleksa Milojević ,&nbsp;Benny Sudakov ,&nbsp;István Tomon","doi":"10.1016/j.jctb.2024.12.009","DOIUrl":"10.1016/j.jctb.2024.12.009","url":null,"abstract":"<div><div>The celebrated Kővári-Sós-Turán theorem states that any <em>n</em>-vertex graph containing no copy of the complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>s</mi></mrow></msub></math></span> has at most <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>s</mi></mrow></msup><mo>)</mo></math></span> edges. In the past two decades, motivated by the applications in discrete geometry and structural graph theory, a number of results demonstrated that this bound can be greatly improved if the graph satisfies certain structural restrictions. We propose the systematic study of this phenomenon, and state the conjecture that if <em>H</em> is a bipartite graph, then an induced <em>H</em>-free and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>s</mi></mrow></msub></math></span>-free graph cannot have much more edges than an <em>H</em>-free graph. We provide evidence for this conjecture by considering trees, cycles, the cube graph, and bipartite graphs with degrees bounded by <em>k</em> on one side, obtaining in all the cases similar bounds as in the non-induced setting. Our results also have applications to the Erdős-Hajnal conjecture, the problem of finding induced <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free subgraphs with large degree and bounding the average degree of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>s</mi></mrow></msub></math></span>-free graphs which do not contain induced subdivisions of a fixed graph.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 168-197"},"PeriodicalIF":1.2,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092868","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":"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":"2025-05-01","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":"The next case of Andrásfai's conjecture","authors":"Tomasz Łuczak ,&nbsp;Joanna Polcyn ,&nbsp;Christian Reiher","doi":"10.1016/j.jctb.2024.12.010","DOIUrl":"10.1016/j.jctb.2024.12.010","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> denote the maximum number of edges in a triangle-free graph on <em>n</em> vertices which contains no independent sets larger than <em>s</em>. The behaviour of <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> was first studied by Andrásfai, who conjectured that for <span><math><mi>s</mi><mo>&gt;</mo><mi>n</mi><mo>/</mo><mn>3</mn></math></span> this function is determined by appropriately chosen blow-ups of so called Andrásfai graphs. Moreover, he proved <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>4</mn><mi>n</mi><mi>s</mi><mo>+</mo><mn>5</mn><msup><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> for <span><math><mi>s</mi><mo>/</mo><mi>n</mi><mo>∈</mo><mo>[</mo><mn>2</mn><mo>/</mo><mn>5</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>]</mo></math></span> and in earlier work we obtained <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>=</mo><mn>3</mn><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>15</mn><mi>n</mi><mi>s</mi><mo>+</mo><mn>20</mn><msup><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> for <span><math><mi>s</mi><mo>/</mo><mi>n</mi><mo>∈</mo><mo>[</mo><mn>3</mn><mo>/</mo><mn>8</mn><mo>,</mo><mn>2</mn><mo>/</mo><mn>5</mn><mo>]</mo></math></span>. Here we make the next step in the quest to settle Andrásfai's conjecture by proving <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>=</mo><mn>6</mn><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>32</mn><mi>n</mi><mi>s</mi><mo>+</mo><mn>44</mn><msup><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> for <span><math><mi>s</mi><mo>/</mo><mi>n</mi><mo>∈</mo><mo>[</mo><mn>4</mn><mo>/</mo><mn>11</mn><mo>,</mo><mn>3</mn><mo>/</mo><mn>8</mn><mo>]</mo></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 198-220"},"PeriodicalIF":1.2,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092869","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":"Toward a density Corrádi–Hajnal theorem for degenerate hypergraphs","authors":"Jianfeng Hou ,&nbsp;Caiyun Hu ,&nbsp;Heng Li ,&nbsp;Xizhi Liu ,&nbsp;Caihong Yang ,&nbsp;Yixiao Zhang","doi":"10.1016/j.jctb.2025.01.001","DOIUrl":"10.1016/j.jctb.2025.01.001","url":null,"abstract":"<div><div>Given an <em>r</em>-graph <em>F</em> with <span><math><mi>r</mi><mo>≥</mo><mn>2</mn></math></span>, let <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>F</mi><mo>)</mo></math></span> denote the maximum number of edges in an <em>n</em>-vertex <em>r</em>-graph with at most <em>t</em> pairwise vertex-disjoint copies of <em>F</em>. Extending several old results and complementing prior work <span><span>[34]</span></span> on nondegenerate hypergraphs, we initiate a systematic study on <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>F</mi><mo>)</mo></math></span> for degenerate hypergraphs <em>F</em>.</div><div>For a broad class of degenerate hypergraphs <em>F</em>, we present near-optimal upper bounds for <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>F</mi><mo>)</mo></math></span> when <em>n</em> is sufficiently large and <em>t</em> lies in intervals <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mi>ε</mi><mo>⋅</mo><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>]</mo></math></span>, <span><math><mo>[</mo><mfrac><mrow><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow><mrow><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>,</mo><mi>ε</mi><mi>n</mi><mo>]</mo></math></span>, and <span><math><mo>[</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>ε</mi><mo>)</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>v</mi><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>v</mi><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mfrac><mo>]</mo></math></span>, where <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span> is a constant depending only on <em>F</em>. Our results reveal very different structures for extremal constructions across the three intervals, and we provide characterizations of extremal constructions within the first interval. Additionally, we characterize extremal constructions within the second interval for graphs. Our proof for the first interval also applies to a special class of nondegenerate hypergraphs, including those with undetermined Turán densities, partially improving a result in <span><span>[34]</span></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 221-262"},"PeriodicalIF":1.2,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092870","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":"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":"2025-05-01","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":"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":"2025-05-01","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":"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":"2025-03-01","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":"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":"2025-03-01","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":"Note on disjoint faces in simple topological graphs","authors":"Ji Zeng","doi":"10.1016/j.jctb.2024.11.002","DOIUrl":"10.1016/j.jctb.2024.11.002","url":null,"abstract":"<div><div>We prove that every <em>n</em>-vertex complete simple topological graph generates at least <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> pairwise disjoint 4-faces. This improves upon a recent result by Hubard and Suk. As an immediate corollary, every <em>n</em>-vertex complete simple topological graph drawn in the unit square generates a 4-face with area at most <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>n</mi><mo>)</mo></math></span>. This can be seen as a topological variant of the Heilbronn problem for quadrilaterals. We construct examples showing that our result is asymptotically tight. We also discuss the similar problem for <em>k</em>-faces with arbitrary <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"171 ","pages":"Pages 28-35"},"PeriodicalIF":1.2,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743022","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 characterization of the Grassmann graphs","authors":"Alexander L. Gavrilyuk ,&nbsp;Jack H. Koolen","doi":"10.1016/j.jctb.2024.11.001","DOIUrl":"10.1016/j.jctb.2024.11.001","url":null,"abstract":"<div><div>The Grassmann graph <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> is a graph on the <em>D</em>-dimensional subspaces of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> with two subspaces being adjacent if their intersection has dimension <span><math><mi>D</mi><mo>−</mo><mn>1</mn></math></span>. Characterizing these graphs by their intersection numbers is an important step towards a solution of the classification problem for <span><math><mo>(</mo><mi>P</mi><mrow><mspace></mspace><mi>and</mi><mspace></mspace></mrow><mi>Q</mi><mo>)</mo></math></span>-polynomial association schemes, posed by Bannai and Ito in their monograph “Algebraic Combinatorics I” (1984).</div><div>Metsch (1995) <span><span>[37]</span></span> showed that the Grassmann graph <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> with <span><math><mi>D</mi><mo>≥</mo><mn>3</mn></math></span> is characterized by its intersection numbers except for the following two principal open cases: <span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>D</mi></math></span> or <span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>D</mi><mo>+</mo><mn>1</mn></math></span>. Van Dam and Koolen (2005) <span><span>[57]</span></span> constructed the twisted Grassmann graphs with the same intersection numbers as the Grassmann graphs <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mn>2</mn><mi>D</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>D</mi><mo>)</mo></math></span> (for any prime power <em>q</em> and <span><math><mi>D</mi><mo>≥</mo><mn>2</mn></math></span>), but not isomorphic to the latter ones. This shows that characterizing the graphs in the remaining cases would require a conceptually new approach.</div><div>We prove that the Grassmann graph <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mn>2</mn><mi>D</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> is characterized by its intersection numbers provided that <em>D</em> is large enough.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"171 ","pages":"Pages 1-27"},"PeriodicalIF":1.2,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142663746","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}