European Journal of Combinatorics最新文献

{"title":"Beyond the pseudoforest strong Nine Dragon Tree Theorem","authors":"Sebastian Mies ,&nbsp;Benjamin Moore ,&nbsp;Evelyne Smith-Roberge","doi":"10.1016/j.ejc.2025.104214","DOIUrl":"10.1016/j.ejc.2025.104214","url":null,"abstract":"<div><div>The pseudoforest version of the Strong Nine Dragon Tree Conjecture states that if a graph <span><math><mi>G</mi></math></span> has maximum average degree <span><math><mrow><mi>m</mi><mi>a</mi><mi>d</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn><msub><mrow><mo>max</mo></mrow><mrow><mi>H</mi><mo>⊆</mo><mi>G</mi></mrow></msub><mfrac><mrow><mi>e</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow><mrow><mi>v</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></mfrac></mrow></math></span> at most <span><math><mrow><mn>2</mn><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow></mrow></math></span>, then it has a decomposition into <span><math><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></math></span> pseudoforests where in one pseudoforest <span><math><mi>F</mi></math></span> the components of <span><math><mi>F</mi></math></span> have at most <span><math><mi>d</mi></math></span> edges. This was proven in 2020 in Grout and Moore (2020). We strengthen this theorem by showing that we can find such a decomposition where additionally <span><math><mi>F</mi></math></span> is acyclic, the diameter of the components of <span><math><mi>F</mi></math></span> is at most <span><math><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>2</mn></mrow></math></span>, where <span><math><mrow><mi>ℓ</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mfenced></mrow></math></span>, and at most <span><math><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></math></span> if <span><math><mrow><mi>d</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>mod</mo><mspace></mspace><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. Furthermore, for any component <span><math><mi>K</mi></math></span> of <span><math><mi>F</mi></math></span> and any <span><math><mrow><mi>z</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, we have <span><math><mrow><mi>d</mi><mi>i</mi><mi>a</mi><mi>m</mi><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><mi>z</mi></mrow></math></span> if <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mo>≥</mo><mi>d</mi><mo>−</mo><mi>z</mi><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span>. We also show that both diameter bounds are best possible as an extension for both the Strong Nine Dragon Tree Conjecture for pseudoforests and its original conjecture for forests. In fact, they are still optimal even if we only enforce <span><math><mi>F</mi></math></span> to have any constant maximum degree, instead of enforcing every component of <span><math><mi>F</mi></math></span> to have at most <span><math><mi>d</mi></math></span> edges.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104214"},"PeriodicalIF":1.0,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"1-planar unit distance graphs","authors":"Panna Gehér ,&nbsp;Géza Tóth","doi":"10.1016/j.ejc.2025.104212","DOIUrl":"10.1016/j.ejc.2025.104212","url":null,"abstract":"<div><div>A matchstick graph is a plane graph with edges drawn as unit distance line segments. This class of graphs was introduced by Harborth who conjectured that a matchstick graph on <span><math><mi>n</mi></math></span> vertices can have at most <span><math><mrow><mo>⌊</mo><mn>3</mn><mi>n</mi><mo>−</mo><msqrt><mrow><mn>12</mn><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msqrt><mo>⌋</mo></mrow></math></span> edges. Recently, his conjecture was settled by Lavollée and Swanepoel. In this paper we consider 1-planar unit distance graphs. We say that a graph is a 1-planar unit distance graph if it can be drawn in the plane such that all edges are drawn as unit distance line segments while each of them are involved in at most one crossing. We show that such graphs on <span><math><mi>n</mi></math></span> vertices can have at most <span><math><mrow><mn>3</mn><mi>n</mi><mo>−</mo><mroot><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></mroot><mo>/</mo><mn>15</mn></mrow></math></span> edges, which is almost tight. We also investigate some generalizations, namely <span><math><mi>k</mi></math></span>-planar and <span><math><mi>k</mi></math></span>-quasiplanar unit distance graphs.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104212"},"PeriodicalIF":1.0,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144549398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Extensions and applications of the Tuza-Vestergaard theorem","authors":"Michael A. Henning ,&nbsp;Anders Yeo","doi":"10.1016/j.ejc.2025.104201","DOIUrl":"10.1016/j.ejc.2025.104201","url":null,"abstract":"&lt;div&gt;&lt;div&gt;The transversal number &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; of a hypergraph &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is the minimum number of vertices that intersect every edge of &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. A 6-uniform hypergraph has all edges of size 6. On 10 November 2000 Tuza and Vestergaard (2002) conjectured that if &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is a 3-regular 6-uniform hypergraph of order &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, then &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&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;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. This conjecture was recently proven by the Henning and Yeo (2023) and is now called the Tuza-Vestergaard Theorem. In this paper we extend the Tuza-Vestergaard Theorem by relaxing the 3-regularity constraint and allowing bounded maximum degree 4. We present several applications of the Tuza-Vestergaard Theorem and its extension. We obtain best known upper bounds to date on the transversal number of a (general) 6-uniform hypergraph &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of order &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and size &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. In particular, if &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is a 4-regular 6-uniform hypergraph of order &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, then we show that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. The Tuza constant &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is defined by &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;sup&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, where the supremum is taken over the class of all 6-uniform hypergraphs &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Since 1990 the exact value of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; has yet to be determined. We show that &lt;span&gt;&lt;math&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;6&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;6&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&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;210&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;6&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is conjectured to be the correct bound. Moreover we show that if &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is a graph of order &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;δ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104201"},"PeriodicalIF":1.0,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144518192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Patterns in multi-dimensional permutations","authors":"Shaoshi Chen ,&nbsp;Hanqian Fang ,&nbsp;Sergey Kitaev ,&nbsp;Candice X.T. Zhang","doi":"10.1016/j.ejc.2025.104203","DOIUrl":"10.1016/j.ejc.2025.104203","url":null,"abstract":"<div><div>In this paper, we propose a general framework that extends the theory of permutation patterns to higher dimensions and unifies several combinatorial objects studied in the literature. Our approach involves introducing the concept of a “level” for an element in a multi-dimensional permutation, which can be defined in multiple ways. We consider two natural definitions of a level, each establishing connections to other combinatorial sequences found in the Online Encyclopedia of Integer Sequences (OEIS).</div><div>Our framework allows us to offer combinatorial interpretations for various sequences found in the OEIS, many of which previously lacked such interpretations. As a notable example, we introduce an elegant combinatorial interpretation for the Springer numbers: they count weakly increasing 3-dimensional permutations under the definition of levels determined by maximal entries.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104203"},"PeriodicalIF":1.0,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144518193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the C-diversity of intersecting k-graphs","authors":"Peter Frankl ,&nbsp;Jian Wang","doi":"10.1016/j.ejc.2025.104199","DOIUrl":"10.1016/j.ejc.2025.104199","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; be a family consisting of &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-subsets of the &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-set &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Suppose that &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is intersecting, i.e., &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mo&gt;0̸&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for all &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; be the maximum degree of &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. For a constant &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; the &lt;span&gt;&lt;math&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;em&gt;-diversity&lt;/em&gt; &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is defined as &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, which was introduced by Magnan, Palmer and Wood recently. Define &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;123&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. It has &lt;span&gt;&lt;math&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-diversity &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. The main result shows that for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;42&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;123&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; with equality if and only if &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is isomorphic to &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;123&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. For the case of ordinary diversity &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104199"},"PeriodicalIF":1.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144338491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Vertex isoperimetry on signed graphs and spectra of non-bipartite Cayley and Cayley sum graphs","authors":"Chunyang Hu,&nbsp;Shiping Liu","doi":"10.1016/j.ejc.2025.104200","DOIUrl":"10.1016/j.ejc.2025.104200","url":null,"abstract":"<div><div>For a non-bipartite finite Cayley graph, we show the non-trivial eigenvalues of its normalized adjacency matrix lie in the interval <span><math><mrow><mfenced><mrow><mo>−</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mi>c</mi><msubsup><mrow><mi>h</mi></mrow><mrow><mi>o</mi><mi>u</mi><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mi>d</mi></mrow></mfrac><mo>,</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mi>C</mi><msubsup><mrow><mi>h</mi></mrow><mrow><mi>o</mi><mi>u</mi><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mi>d</mi></mrow></mfrac></mrow></mfenced><mo>,</mo></mrow></math></span> for some absolute constants <span><math><mi>c</mi></math></span> and <span><math><mi>C</mi></math></span>, where <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>o</mi><mi>u</mi><mi>t</mi></mrow></msub></math></span> stands for the outer vertex boundary isoperimetric constant. This improves upon recent obtained estimates aiming at a quantitative version of a result due to Breuillard, Green, Guralnick and Tao. We achieve this by extending the work of Bobkov, Houdré and Tetali on vertex isoperimetry to the setting of signed graphs. We further extend our interval estimate to the settings of vertex transitive graphs and Cayley sum graphs. As a byproduct, we answer positively open questions proposed recently by Moorman, Ralli and Tetali.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104200"},"PeriodicalIF":1.0,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"10-list recoloring of planar graphs","authors":"Daniel W. Cranston","doi":"10.1016/j.ejc.2025.104190","DOIUrl":"10.1016/j.ejc.2025.104190","url":null,"abstract":"<div><div>Fix a planar graph <span><math><mi>G</mi></math></span> and a list assignment <span><math><mi>L</mi></math></span> with <span><math><mrow><mo>|</mo><mi>L</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>|</mo><mo>=</mo><mn>10</mn></mrow></math></span> for all <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Let <span><math><mi>α</mi></math></span> and <span><math><mi>β</mi></math></span> be <span><math><mi>L</mi></math></span>-colorings of <span><math><mi>G</mi></math></span>. A recoloring sequence from <span><math><mi>α</mi></math></span> to <span><math><mi>β</mi></math></span> is a sequence of <span><math><mi>L</mi></math></span>-colorings, beginning with <span><math><mi>α</mi></math></span> and ending with <span><math><mi>β</mi></math></span>, such that each successive pair in the sequence differs in the color on a single vertex of <span><math><mi>G</mi></math></span>. We show that there exists a constant <span><math><mi>C</mi></math></span> such that for all choices of <span><math><mi>α</mi></math></span> and <span><math><mi>β</mi></math></span> there exists a recoloring sequence <span><math><mi>σ</mi></math></span> from <span><math><mi>α</mi></math></span> to <span><math><mi>β</mi></math></span> that recolors each vertex at most <span><math><mi>C</mi></math></span> times. In particular, <span><math><mi>σ</mi></math></span> has length at most <span><math><mrow><mi>C</mi><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow></math></span>. This confirms a conjecture of Dvořák and Feghali. For our proof, we introduce a new technique for quickly showing that many configurations are reducible. We believe this method may be of independent interest and will have application to other problems in this area.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104190"},"PeriodicalIF":1.0,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144241203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Partitioning 2-edge-coloured bipartite graphs into monochromatic cycles","authors":"Fabrício Siqueira Benevides ,&nbsp;Arthur Lima Quintino ,&nbsp;Alexandre Talon","doi":"10.1016/j.ejc.2025.104192","DOIUrl":"10.1016/j.ejc.2025.104192","url":null,"abstract":"<div><div>Given an <span><math><mi>r</mi></math></span>-edge-colouring of the edges of a graph <span><math><mi>G</mi></math></span>, we say that it can be partitioned into <span><math><mi>p</mi></math></span> monochromatic cycles when there exists a set of <span><math><mi>p</mi></math></span> vertex-disjoint monochromatic cycles covering all the vertices of <span><math><mi>G</mi></math></span>. In the literature of this problem, an edge and a single vertex both count as a cycle.</div><div>We show that for every 2-colouring of the edges of a complete balanced bipartite graph, <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>, it can be partitioned into at most 4 monochromatic cycles. This type of question was first studied in 1970 for complete graphs and in 1983, by Gyárfás and Lehel, for <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>. In 2014, Pokrovskiy, showed for all <span><math><mi>n</mi></math></span> that given any 2-colouring of its edges, <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> can be partitioned into at most three monochromatic paths. It turns out that finding monochromatic cycles instead of paths is a natural question that has also been raised for other graphs. In 2015, Schaudt and Stein showed that 14 cycles are sufficient for sufficiently large 2-edge-coloured <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104192"},"PeriodicalIF":1.0,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144241204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The bunkbed conjecture is not robust to generalisation","authors":"Lawrence Hollom","doi":"10.1016/j.ejc.2025.104188","DOIUrl":"10.1016/j.ejc.2025.104188","url":null,"abstract":"<div><div>The bunkbed conjecture, which has featured in the folklore of probability theory since at least 1985, concerns bond percolation on the product graph <span><math><mrow><mi>G</mi><mo>□</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>. We have two copies <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> of <span><math><mi>G</mi></math></span>, and if <span><math><msup><mrow><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></msup></math></span> and <span><math><msup><mrow><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></math></span> are the copies of a vertex <span><math><mrow><mi>x</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> respectively, then edge <span><math><mrow><msup><mrow><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></msup><msup><mrow><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span> is present. The conjecture states that, for vertices <span><math><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, percolation from <span><math><msup><mrow><mi>u</mi></mrow><mrow><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></msup></math></span> to <span><math><msup><mrow><mi>v</mi></mrow><mrow><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></msup></math></span> is at least as likely as percolation from <span><math><msup><mrow><mi>u</mi></mrow><mrow><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></msup></math></span> to <span><math><msup><mrow><mi>v</mi></mrow><mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></math></span>.</div><div>In this paper we consider three natural generalisations of the bunkbed conjecture; to site percolation, to hypergraphs, and to directed graphs. Our main aim is to show that all these generalisations are false, and to this end we construct a sequence of counterexamples to these statements. However, we also consider under what extra conditions these generalisations might hold, and give some classes of graph for which the bunkbed conjecture for site percolation does hold.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"128 ","pages":"Article 104188"},"PeriodicalIF":1.0,"publicationDate":"2025-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144194538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Moderate deviations of triangle counts in the Erdős-Rényi random graph G(n,m): The lower tail","authors":"José D. Alvarado ,&nbsp;Gabriel Dias do Couto ,&nbsp;Simon Griffiths","doi":"10.1016/j.ejc.2025.104189","DOIUrl":"10.1016/j.ejc.2025.104189","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;△&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; be the number of triangles in a graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. In Goldschmidt et al. (2020) and Neeman et al. (2023) (respectively) the following bounds were proved on the lower tail behaviour of triangle counts in the dense Erdős-Rényi random graphs &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∼&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;: &lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;△&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;δ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;△&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;exp&lt;/mo&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Θ&lt;/mi&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mtext&gt;if&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&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;mn&gt;3&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;≪&lt;/mo&gt;&lt;mi&gt;δ&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;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;exp&lt;/mo&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Θ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mtext&gt;if&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&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;mn&gt;3&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;≪&lt;/mo&gt;&lt;mi&gt;δ&lt;/mi&gt;&lt;mo&gt;≪&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; Neeman et al. (2023) also conjectured that the probability should be of the form &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;exp&lt;/mo&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Θ&lt;/mi&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; in the “missing interval” &lt;span&gt;&lt;math&gt;&lt;mrow&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;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;≪&lt;/mo&gt;&lt;mi&gt;δ&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;mn&gt;3&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. We prove this conjecture. As part of our proof we also prove that some random graph statistics, related to degrees and codegre","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"129 ","pages":"Article 104189"},"PeriodicalIF":1.0,"publicationDate":"2025-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144588327","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}