Journal of Combinatorial Theory Series B最新文献

{"title":"Cubic graphs with no eigenvalues in the interval (−1,1)","authors":"Krystal Guo ,&nbsp;Gordon F. Royle","doi":"10.1016/j.jctb.2025.10.008","DOIUrl":"10.1016/j.jctb.2025.10.008","url":null,"abstract":"<div><div>We give a complete characterization of the cubic graphs with no eigenvalues in the open interval <span><math><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. We first classify the <em>connected</em> cubic graphs with no eigenvalues in <span><math><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> showing that there are two infinite families: one due to Guo and Mohar (2014) <span><span>[7]</span></span> and the other due to Kollár and Sarnak (2021) <span><span>[12]</span></span>, and 13 “sporadic” graphs on at most 32 vertices. Then a not necessarily connected cubic graph has no eigenvalues in <span><math><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> if and only if the same is true for every connected component. This classification allows us to show that <span><math><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> is a maximal spectral gap set for cubic graphs, thereby answering a question of Kollár and Sarnak (2021) <span><span>[12]</span></span>. The techniques used include examination of the small subgraphs that can appear in such a graph and an application of the classification of generalized line graphs.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 561-583"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145442007","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":"2026-01-01","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":"Quantum advantage and CSP complexity","authors":"Lorenzo Ciardo","doi":"10.1016/j.jctb.2025.10.002","DOIUrl":"10.1016/j.jctb.2025.10.002","url":null,"abstract":"<div><div>Information-processing tasks modelled by homomorphisms between relational structures can witness quantum advantage when entanglement is used as a computational resource. We prove that the occurrence of quantum advantage is determined by the same algebraic structure (known as the polymorphism minion) that captures the complexity of CSPs. We investigate the connection between the minion of quantum advantage and other known minions controlling CSP tractability and width. In this way, we make use of complexity results from the algebraic theory of CSPs to characterise the occurrence of quantum advantage in the case of graphs, and to obtain new necessary and sufficient conditions in the case of arbitrary relational structures.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 404-439"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145332382","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":"2026-01-01","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":"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":"2026-01-01","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":"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":"2026-01-01","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":"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":"2026-01-01","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":"Counterexamples to the linkage conjecture for tournaments","authors":"Jia Zhou,&nbsp;Jin Yan","doi":"10.1016/j.jctb.2025.10.005","DOIUrl":"10.1016/j.jctb.2025.10.005","url":null,"abstract":"<div><div>Let <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> be an integer. A digraph <em>D</em> is <em>k</em>-linked if for every set of 2<em>k</em> distinct vertices <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> in <em>D</em>, there exist <em>k</em> pairwise vertex-disjoint paths <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> such that each path <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> starts at <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and ends at <span><math><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> for <span><math><mi>i</mi><mo>∈</mo><mo>[</mo><mi>k</mi><mo>]</mo></math></span>. In 2015, Pokrovskiy conjectured that there exists a function <span><math><mi>g</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> such that every 2<em>k</em>-connected tournament with minimum in-degree and minimum out-degree at least <span><math><mi>g</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> is <em>k</em>-linked in Pokrovskiy (2015) <span><span>[16]</span></span>. In this paper, we disprove Pokrovskiy's conjecture by constructing a family of 2<em>k</em>-connected tournaments of order <span><math><mi>n</mi><mo>≥</mo><mn>14</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> with arbitrarily large minimum semi-degree (depending on <em>n</em>) that are not <em>k</em>-linked. The counterexamples, with sufficiently large order <em>n</em>, also provide a negative answer to the question posed by Girão et al. (2021) <span><span>[8]</span></span>: whether or not 2<em>k</em>-connectivity is sufficient for <em>k</em>-linkage in every tournament with minimum out-degree at least some polynomial in <em>k</em>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 486-497"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145396361","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":"2026-01-01","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":"Induced subgraphs and tree decompositions XVI. Complete bipartite induced minors","authors":"Maria Chudnovsky ,&nbsp;Sepehr Hajebi ,&nbsp;Sophie Spirkl","doi":"10.1016/j.jctb.2025.09.005","DOIUrl":"10.1016/j.jctb.2025.09.005","url":null,"abstract":"<div><div>We prove that for every graph <em>G</em> with a sufficiently large complete bipartite induced minor, either <em>G</em> has an induced minor isomorphic to a large wall, or <em>G</em> contains a large <em>constellation</em>; that is, a complete bipartite induced minor model such that on one side of the bipartition, each branch set is a singleton, and on the other side, each branch set induces a path.</div><div>We further refine this theorem by characterizing the unavoidable induced subgraphs of large constellations as two types of highly structured constellations. These results will be key ingredients in several forthcoming papers of this series.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"176 ","pages":"Pages 287-318"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145265994","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}