Journal of Combinatorial Theory Series B最新文献

{"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}

{"title":"Efficiently distinguishing all tangles in locally finite graphs","authors":"Raphael W. Jacobs,&nbsp;Paul Knappe","doi":"10.1016/j.jctb.2024.03.004","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.03.004","url":null,"abstract":"<div><p>While finite graphs have tree-decompositions that efficiently distinguish all their tangles, locally finite graphs with thick ends need not have such tree-decompositions. We show that every locally finite graph without thick ends admits such a tree-decomposition, in fact a canonical one. Our proof exhibits a thick end at any obstruction to the existence of such tree-decompositions and builds on new methods for the analysis of the limit behaviour of strictly increasing sequences of separations.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"167 ","pages":"Pages 189-214"},"PeriodicalIF":1.4,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000224/pdfft?md5=eb127aa3ba243eafd76779abff8b1e9d&pid=1-s2.0-S0095895624000224-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140188006","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":"Minimum algebraic connectivity and maximum diameter: Aldous–Fill and Guiduli–Mohar conjectures","authors":"Maryam Abdi ,&nbsp;Ebrahim Ghorbani","doi":"10.1016/j.jctb.2024.02.005","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.02.005","url":null,"abstract":"<div><p>Aldous and Fill (2002) conjectured that the maximum relaxation time for the random walk on a connected regular graph with <em>n</em> vertices is <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mfrac><mrow><mn>3</mn><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math></span>. A conjecture by Guiduli and Mohar (1996) predicts the structure of graphs whose algebraic connectivity <em>μ</em> is the smallest among all connected graphs whose minimum degree <em>δ</em> is a given <em>d</em>. We prove that this conjecture implies the Aldous–Fill conjecture for odd <em>d</em>. We pose another conjecture on the structure of <em>d</em>-regular graphs with minimum <em>μ</em>, and show that this also implies the Aldous–Fill conjecture for even <em>d</em>. In the literature, it has been noted empirically that graphs with small <em>μ</em> tend to have a large diameter. In this regard, Guiduli (1996) asked if the cubic graphs with maximum diameter have algebraic connectivity smaller than all others. Motivated by these, we investigate the interplay between the graphs with maximum diameter and those with minimum algebraic connectivity. We show that the answer to Guiduli problem in its general form, that is for <em>d</em>-regular graphs for every <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> is negative. We aim to develop an asymptotic formulation of the problem. It is proven that <em>d</em>-regular graphs for <span><math><mi>d</mi><mo>≥</mo><mn>5</mn></math></span> as well as graphs with <span><math><mi>δ</mi><mo>=</mo><mi>d</mi></math></span> for <span><math><mi>d</mi><mo>≥</mo><mn>4</mn></math></span> with asymptotically maximum diameter, do not necessarily exhibit the asymptotically smallest <em>μ</em>. We conjecture that <em>d</em>-regular graphs (or graphs with <span><math><mi>δ</mi><mo>=</mo><mi>d</mi></math></span>) that have asymptotically smallest <em>μ</em>, should have asymptotically maximum diameter. The above results rely heavily on our understanding of the structure as well as optimal estimation of the algebraic connectivity of nearly maximum-diameter graphs, from which the Aldous–Fill conjecture for this family of graphs also follows.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"167 ","pages":"Pages 164-188"},"PeriodicalIF":1.4,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140162655","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}