Journal of Combinatorial Theory Series B最新文献

{"title":"Chromatic number is not tournament-local","authors":"António Girão ,&nbsp;Kevin Hendrey ,&nbsp;Freddie Illingworth ,&nbsp;Florian Lehner ,&nbsp;Lukas Michel ,&nbsp;Michael Savery ,&nbsp;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 ,&nbsp;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 ,&nbsp;Jie Ma ,&nbsp;Sergey Norin ,&nbsp;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}

{"title":"Mutual embeddability in groups, trees, and spheres","authors":"Claude Tardif","doi":"10.1016/j.jctb.2024.04.002","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.04.002","url":null,"abstract":"<div><p>Two subsets in a group are called <em>twins</em> if each is contained in a left translate of the other, though the two sets themselves are not translates of each other. We show that in the free group <span><math><msub><mrow><mi>F</mi></mrow><mrow><mo>{</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>}</mo></mrow></msub></math></span>, there exist maximal families of twins of any finite cardinality. This result is used to show that in the context of embeddings of trees, there exist maximal families of twin trees of any finite cardinality. These are counterexamples to the “tree alternative” conjecture, which supplement the first counterexamples published by Kalow, Laflamme, Tateno, and Woodrow. We also investigate twin sets in the sphere <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, where the embeddings considered are isometries of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. We show that there exist maximal families of twin sets in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> of any finite cardinality.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"168 ","pages":"Pages 1-10"},"PeriodicalIF":1.4,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140651035","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":"Random homomorphisms into the orthogonality graph","authors":"Dávid Kunszenti-Kovács,&nbsp;László Lovász,&nbsp;Balázs Szegedy","doi":"10.1016/j.jctb.2024.03.007","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.03.007","url":null,"abstract":"<div><p>Subgraph densities have been defined, and served as basic tools, both in the case of graphons (limits of dense graph sequences) and graphings (limits of bounded-degree graph sequences). While limit objects have been described for the “middle ranges”, the notion of subgraph densities in these limit objects remains elusive. We define subgraph densities in the orthogonality graphs on the unit spheres in dimension <em>d</em>, under an appropriate sparsity condition on the subgraphs. These orthogonality graphs exhibit the main difficulties of defining subgraphs in the “middle” range, and so we expect their study to serve as a key example to defining subgraph densities in more general Markov spaces. Interest in studying homomorphisms of a finite graph <em>G</em> into orthogonality graphs is supported by the fact that such homomorphisms are just the orthonormal representations of the complementary graph.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"167 ","pages":"Pages 392-444"},"PeriodicalIF":1.4,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S009589562400025X/pdfft?md5=28788fda48a1c1b846f39294ec87eb1b&pid=1-s2.0-S009589562400025X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140557933","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":"Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure","authors":"Clément Dallard ,&nbsp;Martin Milanič ,&nbsp;Kenny Štorgel","doi":"10.1016/j.jctb.2024.03.005","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.03.005","url":null,"abstract":"<div><p>We continue the study of <span><math><mo>(</mo><mrow><mi>tw</mi></mrow><mo>,</mo><mi>ω</mi><mo>)</mo></math></span>-bounded graph classes, that is, hereditary graph classes in which the treewidth can only be large due to the presence of a large clique, with the goal of understanding the extent to which this property has useful algorithmic implications for the Maximum Independent Set and related problems.</p><p>In the previous paper of the series [Dallard, Milanič, and Štorgel, Treewidth versus clique number. II. Tree-independence number, J. Comb. Theory, Ser. B, 164 (2024) 404–442], we introduced the <em>tree-independence number</em>, a min-max graph invariant related to tree decompositions. Bounded tree-independence number implies both <span><math><mo>(</mo><mrow><mi>tw</mi></mrow><mo>,</mo><mi>ω</mi><mo>)</mo></math></span>-boundedness and the existence of a polynomial-time algorithm for the Maximum Weight Independent Packing problem, provided that the input graph is given together with a tree decomposition with bounded independence number. In particular, this implies polynomial-time solvability of the Maximum Weight Independent Set problem.</p><p>In this paper, we consider six graph containment relations—the subgraph, topological minor, and minor relations, as well as their induced variants—and for each of them characterize the graphs <em>H</em> for which any graph excluding <em>H</em> with respect to the relation admits a tree decomposition with bounded independence number. The induced minor relation is of particular interest: we show that excluding either a <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> minus an edge or the 4-wheel implies the existence of a tree decomposition in which every bag is a clique plus at most 3 vertices, while excluding a complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>q</mi></mrow></msub></math></span> implies the existence of a tree decomposition with independence number at most <span><math><mn>2</mn><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>.</p><p>These results are obtained using a variety of tools, including <em>ℓ</em>-refined tree decompositions, SPQR trees, and potential maximal cliques, and actually show that in the bounded cases identified in this work, one can also compute tree decompositions with bounded independence number efficiently. Applying the algorithmic framework provided by the previous paper in the series leads to polynomial-time algorithms for the Maximum Weight Independent Set problem in an infinite family of graph classes, each of which properly contains the class of chordal graphs. In particular, these results apply to the class of 1-perfectly orientable graphs, answering a question of Beisegel, Chudnovsky, Gurvich, Milanič, and Servatius from 2019.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"167 ","pages":"Pages 338-391"},"PeriodicalIF":1.4,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000236/pdfft?md5=6e6e7de618af9c521f4bf056c84fcb7b&pid=1-s2.0-S0095895624000236-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140540111","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":"Cancellative hypergraphs and Steiner triple systems","authors":"Xizhi Liu","doi":"10.1016/j.jctb.2024.03.006","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.03.006","url":null,"abstract":"<div><p>A triple system is cancellative if it does not contain three distinct sets <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></math></span> such that the symmetric difference of <em>A</em> and <em>B</em> is contained in <em>C</em>. We show that every cancellative triple system <span><math><mi>H</mi></math></span> that satisfies a particular inequality between the sizes of <span><math><mi>H</mi></math></span> and its shadow must be structurally close to the balanced blowup of some Steiner triple system. Our result contains a stability theorem for cancellative triple systems due to Keevash and Mubayi as a special case. It also implies that the boundary of the feasible region of cancellative triple systems has infinitely many local maxima, thus giving the first example showing this phenomenon.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"167 ","pages":"Pages 303-337"},"PeriodicalIF":1.4,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140341140","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":"Typical structure of hereditary properties of binary matroids","authors":"Stefan Grosser ,&nbsp;Hamed Hatami ,&nbsp;Peter Nelson ,&nbsp;Sergey Norin","doi":"10.1016/j.jctb.2024.03.001","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.03.001","url":null,"abstract":"<div><p>We prove an arithmetic analogue of the typical structure theorem for graph hereditary properties due to Alon, Balogh, Bollobás and Morris <span>[2]</span>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"167 ","pages":"Pages 283-302"},"PeriodicalIF":1.4,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140309525","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":"Sparse graphs with bounded induced cycle packing number have logarithmic treewidth","authors":"Marthe Bonamy ,&nbsp;Édouard Bonnet ,&nbsp;Hugues Déprés ,&nbsp;Louis Esperet ,&nbsp;Colin Geniet ,&nbsp;Claire Hilaire ,&nbsp;Stéphan Thomassé ,&nbsp;Alexandra Wesolek","doi":"10.1016/j.jctb.2024.03.003","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.03.003","url":null,"abstract":"<div><p>A graph is <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-free if it does not contain <em>k</em> pairwise vertex-disjoint and non-adjacent cycles. We prove that “sparse” (here, not containing large complete bipartite graphs as subgraphs) <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is optimal, as there is an infinite family of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-free graphs without <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>3</mn></mrow></msub></math></span> as a subgraph and whose treewidth is (at least) logarithmic.</p><p>Using our result, we show that <span>Maximum Independent Set</span> and <span>3-Coloring</span> in <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-free graphs can be solved in quasi-polynomial time. Other consequences include that most of the central NP-complete problems (such as <span>Maximum Independent Set</span>, <span>Minimum Vertex Cover</span>, <span>Minimum Dominating Set</span>, <span>Minimum Coloring</span>) can be solved in polynomial time in sparse <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-free graphs, and that deciding the <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-freeness of sparse graphs is polynomial time solvable.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"167 ","pages":"Pages 215-249"},"PeriodicalIF":1.4,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140296899","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":"Percolation on dense random graphs with given degrees","authors":"Lyuben Lichev ,&nbsp;Dieter Mitsche ,&nbsp;Guillem Perarnau","doi":"10.1016/j.jctb.2024.03.002","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.03.002","url":null,"abstract":"<div><p>In this paper, we study the order of the largest connected component of a random graph having two sources of randomness: first, the graph is chosen randomly from all graphs with a given degree sequence, and then bond percolation is applied. Far from being able to classify all such degree sequences, we exhibit several new threshold phenomena for the order of the largest component in terms of both sources of randomness. We also provide an example of a degree sequence for which the order of the largest component undergoes an unbounded number of jumps in terms of the percolation parameter, giving rise to a behavior that cannot be observed without percolation.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"167 ","pages":"Pages 250-282"},"PeriodicalIF":1.4,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140296900","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}