{"title":"Efficiently distinguishing all tangles in locally finite graphs","authors":"Raphael W. Jacobs, 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 , 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}
{"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}
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