Journal of Combinatorial Theory Series B最新文献

{"title":"The Hamilton space of pseudorandom graphs","authors":"Micha Christoph ,&nbsp;Rajko Nenadov ,&nbsp;Kalina Petrova","doi":"10.1016/j.jctb.2025.09.002","DOIUrl":"10.1016/j.jctb.2025.09.002","url":null,"abstract":"<div><div>We show that if <em>n</em> is odd and <span><math><mi>p</mi><mo>≥</mo><mi>C</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>/</mo><mi>n</mi></math></span>, then with high probability Hamilton cycles in <span><math><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> span its cycle space. More generally, we show this holds for a class of graphs satisfying certain natural pseudorandom properties. The proof is based on a novel idea of parity-switchers, which can be thought of as analogues of absorbers in the context of cycle spaces. As another application of our method, we show that Hamilton cycles in a near-Dirac graph <em>G</em>, that is, a graph <em>G</em> with odd <em>n</em> vertices and minimum degree <span><math><mi>n</mi><mo>/</mo><mn>2</mn><mo>+</mo><mi>C</mi></math></span> for sufficiently large constant <em>C</em>, span its cycle space.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 254-267"},"PeriodicalIF":1.2,"publicationDate":"2025-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145157540","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":"Local properties of the spectral radius and Perron vector in graphs","authors":"Lele Liu ,&nbsp;Bo Ning","doi":"10.1016/j.jctb.2025.09.001","DOIUrl":"10.1016/j.jctb.2025.09.001","url":null,"abstract":"<div><div>In 2002, Nikiforov proved that for an <em>n</em>-vertex graph <em>G</em> with clique number <em>ω</em> and edge number <em>m</em>, its spectral radius <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> satisfies <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><msqrt><mrow><mn>2</mn><mo>(</mo><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>ω</mi><mo>)</mo><mi>m</mi></mrow></msqrt></math></span>, which confirmed a conjecture implicitly suggested by Edwards and Elphick. In this paper, we prove a local version of spectral Turán inequality, showing that <span><math><msup><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>2</mn><msub><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mfrac><mrow><mi>c</mi><mo>(</mo><mi>e</mi><mo>)</mo><mo>−</mo><mn>1</mn></mrow><mrow><mi>c</mi><mo>(</mo><mi>e</mi><mo>)</mo></mrow></mfrac></math></span>, where <span><math><mi>c</mi><mo>(</mo><mi>e</mi><mo>)</mo></math></span> is the order of the largest clique containing the edge <em>e</em> in <em>G</em>. We also characterize the extremal graphs. Furthermore, we prove that our theorem implies Nikiforov's theorem and provide an example in which the difference of Nikiforov's bound and ours is <span><math><mi>Ω</mi><mo>(</mo><msqrt><mrow><mi>m</mi></mrow></msqrt><mo>)</mo></math></span> for some cases. Our second result explores local properties of the Perron vector of graphs. We disprove a conjecture of Gregory, asserting that for a connected <em>n</em>-vertex graph <em>G</em> with chromatic number <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and an independent set <em>S</em>, we have<span><span><span><math><munder><mo>∑</mo><mrow><mi>v</mi><mo>∈</mo><mi>S</mi></mrow></munder><msubsup><mrow><mi>x</mi></mrow><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow><mrow><mn>2</mn><msqrt><mrow><msup><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>4</mn><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msqrt></mrow></mfrac><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>v</mi></mrow></msub></math></span> is the component of the Perron vector of <em>G</em> with respect to the vertex <em>v</em>. A modified version of Gregory's conjecture is proposed.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 241-253"},"PeriodicalIF":1.2,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145157539","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":"Detachable pairs in 3-connected matroids and simple 3-connected graphs","authors":"Nick Brettell ,&nbsp;Charles Semple ,&nbsp;Gerry Toft","doi":"10.1016/j.jctb.2025.09.003","DOIUrl":"10.1016/j.jctb.2025.09.003","url":null,"abstract":"<div><div>Let <em>M</em> be a 3-connected matroid. A pair <span><math><mo>{</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo>}</mo></math></span> in <em>M</em> is <em>detachable</em> if <span><math><mi>M</mi><mo>﹨</mo><mi>e</mi><mo>﹨</mo><mi>f</mi></math></span> or <span><math><mi>M</mi><mo>/</mo><mi>e</mi><mo>/</mo><mi>f</mi></math></span> is 3-connected. Williams (2015) proved that if <em>M</em> has at least 13 elements, then at least one of the following holds: <em>M</em> has a detachable pair, <em>M</em> has a 3-element circuit or cocircuit, or <em>M</em> is a spike. We address the case where <em>M</em> has a 3-element circuit or cocircuit, to obtain a characterisation of when a matroid with at least 13 elements has a detachable pair. As a consequence, we characterise when a simple 3-connected graph <em>G</em> with <span><math><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mn>13</mn></math></span> has a pair of edges <span><math><mo>{</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo>}</mo></math></span> such that <span><math><mi>G</mi><mo>/</mo><mi>e</mi><mo>/</mo><mi>f</mi></math></span> or <span><math><mi>G</mi><mo>﹨</mo><mi>e</mi><mo>﹨</mo><mi>f</mi></math></span> is simple and 3-connected.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 163-240"},"PeriodicalIF":1.2,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145121193","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 degree-restricted random process is far from uniform","authors":"Michael Molloy ,&nbsp;Erlang Surya ,&nbsp;Lutz Warnke","doi":"10.1016/j.jctb.2025.08.001","DOIUrl":"10.1016/j.jctb.2025.08.001","url":null,"abstract":"<div><div>The degree-restricted random process is a natural algorithmic model for generating graphs with degree sequence <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span>: starting with an empty <em>n</em>-vertex graph, it sequentially adds new random edges so that the degree of each vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> remains at most <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. Wormald conjectured in 1999 that, for <em>d</em>-regular degree sequences <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the final graph of this process is similar to a uniform random <em>d</em>-regular graph.</div><div>In this paper we show that, for degree sequences <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> that are not nearly regular, the final graph of the degree-restricted random process differs substantially from a uniform random graph with degree sequence <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. The combinatorial proof technique is our main conceptual contribution: we adapt the switching method to the degree-restricted process, demonstrating that this enumeration technique can also be used to analyze stochastic processes (rather than just uniform random models, as before).</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 111-162"},"PeriodicalIF":1.2,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145121194","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":"Dense minors of graphs with independence number two","authors":"Sergey Norin ,&nbsp;Paul Seymour","doi":"10.1016/j.jctb.2025.08.005","DOIUrl":"10.1016/j.jctb.2025.08.005","url":null,"abstract":"<div><div>Motivated by Hadwiger's conjecture, we prove that every graph with no independent set of size three contains a <em>t</em>-vertex simple minor with<span><span><span><math><mn>0.98688</mn><mo>⋅</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>t</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mi>o</mi><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span></span></span> edges, where <em>t</em> is its chromatic number.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 101-110"},"PeriodicalIF":1.2,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145105388","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":"Thomassen's theorem on the two-linkage problem in acyclic digraphs: A shorter proof","authors":"Paul Seymour","doi":"10.1016/j.jctb.2025.08.006","DOIUrl":"10.1016/j.jctb.2025.08.006","url":null,"abstract":"<div><div>Let <em>G</em> be an acyclic digraph, and let <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, where <span><math><mi>a</mi><mo>,</mo><mi>b</mi></math></span> are sources, <span><math><mi>c</mi><mo>,</mo><mi>d</mi></math></span> are sinks, and every other vertex has in-degree and out-degree at least two. In 1985, Thomassen showed that there do not exist disjoint directed paths from <em>a</em> to <em>c</em> and from <em>b</em> to <em>d</em>, if and only if <em>G</em> can be drawn in a closed disc with <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi></math></span> drawn in the boundary in order. We give a shorter proof.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 97-100"},"PeriodicalIF":1.2,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049038","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":"Tree independence number II. Three-path-configurations","authors":"Maria Chudnovsky ,&nbsp;Sepehr Hajebi ,&nbsp;Daniel Lokshtanov ,&nbsp;Sophie Spirkl","doi":"10.1016/j.jctb.2025.08.003","DOIUrl":"10.1016/j.jctb.2025.08.003","url":null,"abstract":"<div><div>A <em>three-path-configuration</em> is a graph consisting of three pairwise internally-disjoint paths the union of every two of which is an induced cycle of length at least four. A graph is <em>3PC-free</em> if no induced subgraph of it is a three-path-configuration. We prove that 3PC-free graphs have poly-logarithmic tree independence number. More explicitly, we show that there exists a constant <em>c</em> such that every <em>n</em>-vertex 3PC-free graph has a tree decomposition in which every bag has stability number at most <span><math><mi>c</mi><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span>. This implies that the <span>Maximum Weight Independent Set</span> problem, as well as several other natural algorithmic problems, that are known to be <span>NP</span>-hard in general, can be solved in quasi-polynomial time if the input graph is 3PC-free.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 74-96"},"PeriodicalIF":1.2,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010349","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":"Reuniting χ-boundedness with polynomial χ-boundedness","authors":"Maria Chudnovsky ,&nbsp;Linda Cook ,&nbsp;James Davies ,&nbsp;Sang-il Oum","doi":"10.1016/j.jctb.2025.08.002","DOIUrl":"10.1016/j.jctb.2025.08.002","url":null,"abstract":"<div><div>A class <span><math><mi>F</mi></math></span> of graphs is <em>χ</em>-bounded if there is a function <em>f</em> such that <span><math><mi>χ</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mi>f</mi><mo>(</mo><mi>ω</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>)</mo></math></span> for all induced subgraphs <em>H</em> of a graph in <span><math><mi>F</mi></math></span>. If <em>f</em> can be chosen to be a polynomial, we say that <span><math><mi>F</mi></math></span> is polynomially <em>χ</em>-bounded. Esperet proposed a conjecture that every <em>χ</em>-bounded class of graphs is polynomially <em>χ</em>-bounded. This conjecture has been disproved; it has been shown that there are classes of graphs that are <em>χ</em>-bounded but not polynomially <em>χ</em>-bounded. Nevertheless, inspired by Esperet's conjecture, we introduce Pollyanna classes of graphs. A class <span><math><mi>C</mi></math></span> of graphs is Pollyanna if <span><math><mi>C</mi><mo>∩</mo><mi>F</mi></math></span> is polynomially <em>χ</em>-bounded for every <em>χ</em>-bounded class <span><math><mi>F</mi></math></span> of graphs. We prove that several classes of graphs are Pollyanna and also present some proper classes of graphs that are not Pollyanna.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 30-73"},"PeriodicalIF":1.2,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144907692","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 graphs without cycles of length 0 modulo 4","authors":"Ervin Győri ,&nbsp;Binlong Li ,&nbsp;Nika Salia ,&nbsp;Casey Tompkins ,&nbsp;Kitti Varga ,&nbsp;Manran Zhu","doi":"10.1016/j.jctb.2025.07.008","DOIUrl":"10.1016/j.jctb.2025.07.008","url":null,"abstract":"<div><div>Bollobás proved that for every <em>k</em> and <em>ℓ</em> such that <span><math><mi>k</mi><mi>Z</mi><mo>+</mo><mi>ℓ</mi></math></span> contains an even number, an <em>n</em>-vertex graph containing no cycle of length <span><math><mi>ℓ</mi><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>k</mi></math></span> can contain at most a linear number of edges. The precise (or asymptotic) value of the maximum number of edges in such a graph is known for very few pairs <em>ℓ</em> and <em>k</em>. In this work we precisely determine the maximum number of edges in a graph containing no cycle of length <span><math><mn>0</mn><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 7-29"},"PeriodicalIF":1.2,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144863500","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":"Closure property of contraction-depth of matroids","authors":"Marcin Briański, Daniel Král', Ander Lamaison","doi":"10.1016/j.jctb.2025.07.006","DOIUrl":"https://doi.org/10.1016/j.jctb.2025.07.006","url":null,"abstract":"Contraction<ce:sup loc=\"post\">⁎</ce:sup>-depth is a matroid depth parameter analogous to tree-depth of graphs. We establish the matroid analogue of the classical graph theory result asserting that the tree-depth of a graph <ce:italic>G</ce:italic> is the minimum height of a rooted forest whose closure contains <ce:italic>G</ce:italic> by proving the following for every matroid <ce:italic>M</ce:italic> (except the trivial case when <ce:italic>M</ce:italic> consists of loops and coloops only): the contraction<ce:sup loc=\"post\">⁎</ce:sup>-depth of <ce:italic>M</ce:italic> plus one is equal to the minimum contraction-depth of a matroid containing <ce:italic>M</ce:italic> as a restriction.","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"53 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2025-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144900112","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}