Journal of Combinatorial Theory Series B最新文献

{"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 ,&nbsp;Simón Piga ,&nbsp;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":"&lt;div&gt;&lt;p&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be a family of &lt;em&gt;r&lt;/em&gt;-uniform hypergraphs. The random Turán number &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;ex&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is the maximum number of edges in an &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-free subgraph of &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; is the Erdős-Rényi random &lt;em&gt;r&lt;/em&gt;-graph with parameter &lt;em&gt;p&lt;/em&gt;. Let &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; denote the &lt;em&gt;r&lt;/em&gt;-uniform linear cycle of length &lt;em&gt;ℓ&lt;/em&gt;. For &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, Mubayi and Yepremyan showed that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;ex&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;max&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. This upper bound is not tight when &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, we close the gap for &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. More precisely, we show that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;ex&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;Θ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; when &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;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}

{"title":"The maximum number of copies of an even cycle in a planar graph","authors":"Zequn Lv ,&nbsp;Ervin Győri ,&nbsp;Zhen He ,&nbsp;Nika Salia ,&nbsp;Casey Tompkins ,&nbsp;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 ,&nbsp;Hong Liu ,&nbsp;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,&nbsp;Zhihan Jin,&nbsp;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>&gt;</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>&gt;</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}

{"title":"The core conjecture of Hilton and Zhao","authors":"Yan Cao ,&nbsp;Guantao Chen ,&nbsp;Guangming Jing ,&nbsp;Songling Shan","doi":"10.1016/j.jctb.2024.01.004","DOIUrl":"10.1016/j.jctb.2024.01.004","url":null,"abstract":"<div><p>A simple graph <em>G</em><span> with maximum degree Δ is </span><em>overfull</em> if <span><math><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>&gt;</mo><mi>Δ</mi><mo>⌊</mo><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>/</mo><mn>2</mn><mo>⌋</mo></math></span>. The <em>core</em> of <em>G</em>, denoted <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>Δ</mi></mrow></msub></math></span>, is the subgraph of <em>G</em> induced by its vertices of degree Δ. Clearly, the chromatic index of <em>G</em> equals <span><math><mi>Δ</mi><mo>+</mo><mn>1</mn></math></span> if <em>G</em> is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if <em>G</em><span> is a simple connected graph with </span><span><math><mi>Δ</mi><mo>≥</mo><mn>3</mn></math></span> and <span><math><mi>Δ</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>Δ</mi></mrow></msub><mo>)</mo><mo>≤</mo><mn>2</mn></math></span>, then <span><math><msup><mrow><mi>χ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Δ</mi><mo>+</mo><mn>1</mn></math></span> implies that <em>G</em> is overfull or <span><math><mi>G</mi><mo>=</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, where <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is obtained from the Petersen graph by deleting a vertex. Cariolaro and Cariolaro settled the base case <span><math><mi>Δ</mi><mo>=</mo><mn>3</mn></math></span> in 2003, and Cranston and Rabern proved the next case, <span><math><mi>Δ</mi><mo>=</mo><mn>4</mn></math></span>, in 2019. In this paper, we give a proof of this conjecture for all <span><math><mi>Δ</mi><mo>≥</mo><mn>4</mn></math></span>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"166 ","pages":"Pages 154-182"},"PeriodicalIF":1.4,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139551178","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 orders of automorphisms of vertex-transitive graphs","authors":"Primož Potočnik ,&nbsp;Micael Toledo ,&nbsp;Gabriel Verret","doi":"10.1016/j.jctb.2024.01.001","DOIUrl":"10.1016/j.jctb.2024.01.001","url":null,"abstract":"<div><p>In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with <em>n</em> vertices and of valence <em>d</em>, <span><math><mi>d</mi><mo>≤</mo><mn>4</mn></math></span>, is at most <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msub><mi>n</mi></math></span> where <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span> and <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>=</mo><mn>9</mn></math></span>. Whether such a constant <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> exists for valencies larger than 4 remains an unanswered question. Further, we prove that every automorphism <em>g</em> of a finite connected 3-valent vertex-transitive graph Γ, <span><math><mi>Γ</mi><mo>≇</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>3</mn></mrow></msub></math></span>, has a regular orbit, that is, an orbit of <span><math><mo>〈</mo><mi>g</mi><mo>〉</mo></math></span> of length equal to the order of <em>g</em>. Moreover, we prove that in this case either Γ belongs to a well understood family of exceptional graphs or at least 5/12 of the vertices of Γ belong to a regular orbit of <em>g</em>. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms <em>C</em> of a connected 3-valent vertex-transitive graph Γ in terms of the number of vertices of Γ and the length of a longest orbit of <em>C</em>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"166 ","pages":"Pages 123-153"},"PeriodicalIF":1.4,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000029/pdfft?md5=5bdead4227eef1a873304cc296bf7df1&pid=1-s2.0-S0095895624000029-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139522954","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":"Extended commonality of paths and cycles via Schur convexity","authors":"Jang Soo Kim ,&nbsp;Joonkyung Lee","doi":"10.1016/j.jctb.2023.12.001","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.12.001","url":null,"abstract":"<div><p>A graph <em>H</em> is <em>common</em> if the number of monochromatic copies of <em>H</em> in a 2-edge-colouring of the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is asymptotically minimised by the random colouring, or equivalently, <span><math><msub><mrow><mi>t</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mi>W</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>−</mo><mi>W</mi><mo>)</mo><mo>≥</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>1</mn><mo>−</mo><mi>e</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></msup></math></span> holds for every graphon <span><math><mi>W</mi><mo>:</mo><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, where <span><math><msub><mrow><mi>t</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mo>.</mo><mo>)</mo></math></span><span> denotes the homomorphism density of the graph </span><em>H</em>. Paths and cycles being common is one of the earliest cornerstones in extremal graph theory, due to Mulholland and Smith (1959), Goodman (1959), and Sidorenko (1989).</p><p>We prove a graph homomorphism inequality that extends the commonality of paths and cycles. Namely, <span><math><msub><mrow><mi>t</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mi>W</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>−</mo><mi>W</mi><mo>)</mo><mo>≥</mo><msub><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><msup><mrow><mo>(</mo><mi>W</mi><mo>)</mo></mrow><mrow><mi>e</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></msup><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>W</mi><mo>)</mo></mrow><mrow><mi>e</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></msup></math></span> whenever <em>H</em> is a path or a cycle and <span><math><mi>W</mi><mo>:</mo><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>→</mo><mi>R</mi></math></span><span> is a bounded symmetric measurable function.</span></p><p>This answers a question of Sidorenko from 1989, who proved a slightly weaker result for even-length paths to prove the commonality of odd cycles. Furthermore, it also settles a recent conjecture of Behague, Morrison, and Noel in a strong form, who asked if the inequality holds for graphons <em>W</em> and odd cycles <em>H</em><span>. Our proof uses Schur convexity of complete homogeneous symmetric functions, which may be of independent interest.</span></p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"166 ","pages":"Pages 109-122"},"PeriodicalIF":1.4,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139487840","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}