European Journal of Combinatorics最新文献

{"title":"Polynomial expressions for the dimensions of the representations of symmetric groups and restricted standard Young tableaux","authors":"Avichai Cohen,&nbsp;Shaul Zemel","doi":"10.1016/j.ejc.2025.104242","DOIUrl":"10.1016/j.ejc.2025.104242","url":null,"abstract":"<div><div>Given a partition <span><math><mi>λ</mi></math></span> of a number <span><math><mi>k</mi></math></span>, it is known that by adding a long line of length <span><math><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></math></span>, the dimension of the associated representation of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is an integer-valued polynomial of degree <span><math><mi>k</mi></math></span> in <span><math><mi>n</mi></math></span>. We show that its expansion in the binomial basis is bounded by the length of <span><math><mi>λ</mi></math></span>, and that the resulting coefficient of index <span><math><mi>h</mi></math></span>, with alternating signs, counts the standard Young tableaux of shape <span><math><mi>λ</mi></math></span> in which a given collection of consecutive <span><math><mi>h</mi></math></span> numbers lie in increasing rows. We also construct bijections in order to demonstrate explicitly that this number is indeed independent of the set of consecutive <span><math><mi>h</mi></math></span> numbers used.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"131 ","pages":"Article 104242"},"PeriodicalIF":0.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096500","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 exact Turán number of disjoint graphs– A generalization of Simonovits’ theorem, and beyond","authors":"Guantao Chen ,&nbsp;Xingyu Lei ,&nbsp;Shuchao Li","doi":"10.1016/j.ejc.2025.104226","DOIUrl":"10.1016/j.ejc.2025.104226","url":null,"abstract":"&lt;div&gt;&lt;div&gt;For a given graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, we say that a graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;em&gt;-free&lt;/em&gt; if it does not contain &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; as a subgraph. Let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mtext&gt;ex&lt;/mtext&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;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; (resp. &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mtext&gt;ex&lt;/mtext&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;) denote the maximum size (resp. spectral radius) of an &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-vertex &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-free graph, and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mtext&gt;Ex&lt;/mtext&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;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; (resp. &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mtext&gt;Ex&lt;/mtext&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;) denote the set of all &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-vertex &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-free graphs with &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mtext&gt;ex&lt;/mtext&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;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; edges (resp. spectral radius &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mtext&gt;ex&lt;/mtext&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;). We call &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mtext&gt;ex&lt;/mtext&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;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; (resp. &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mtext&gt;ex&lt;/mtext&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;H&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;) the &lt;em&gt;Turán number&lt;/em&gt; (resp. &lt;em&gt;spectral Turán number&lt;/em&gt;) of &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Suppose that we know the exact values of Turán numbers of &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;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&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;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, respectively. Can we get the exact value of the Turán number of the disjoint union of &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;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∪&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&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;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;? Moon considered the disjoint union of complete graphs. A graph &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is &lt;em&gt;color-critical&lt;/em&gt; if there exists an edge &lt;span&gt;&lt;math&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;χ&lt;/mi&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;. Simonovits extended Moon’s result to the disjoint union of &lt;em&gt;color-critical graphs&lt;/em&gt; for sufficiently large &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Erdős et al. determined the Turán number of triangles sha","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104226"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144772968","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":"Generalized diagonals in positive semi-definite matrices","authors":"Robert Angarone ,&nbsp;Daniel Soskin","doi":"10.1016/j.ejc.2025.104220","DOIUrl":"10.1016/j.ejc.2025.104220","url":null,"abstract":"<div><div>We describe all inequalities among generalized diagonals in positive semi-definite matrices. These turn out to be governed by a simple partial order on the symmetric group. This provides an analogue of results of Drake, Gerrish, and Skandera on inequalities among generalized diagonals in totally nonnegative matrices.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104220"},"PeriodicalIF":1.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144611621","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":"2025-12-01","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":"Quasisymmetric Schur Q-functions and peak Young quasisymmetric Schur functions","authors":"Seung-Il Choi ,&nbsp;Sun-Young Nam ,&nbsp;Young-Tak Oh","doi":"10.1016/j.ejc.2025.104213","DOIUrl":"10.1016/j.ejc.2025.104213","url":null,"abstract":"<div><div>In this paper, we explore the relationship between quasisymmetric Schur <span><math><mi>Q</mi></math></span>-functions and peak Young quasisymmetric Schur functions. We introduce a bijection on <span><math><mrow><mi>SPIT</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span> such that <span><math><mrow><mo>{</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>∣</mo><mi>T</mi><mo>∈</mo><mi>SPIT</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>}</mo></mrow></math></span> and <span><math><mrow><mo>{</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>∣</mo><mi>T</mi><mo>∈</mo><mi>SPIT</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>}</mo></mrow></math></span> share identical descent distributions. Here, <span><math><mrow><mi>SPIT</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span> is the set of standard peak immaculate tableaux of shape <span><math><mi>α</mi></math></span>, and <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> denote column reading and row reading, respectively. By combining this equidistribution with the algorithm developed by Allen, Hallam, and Mason, we demonstrate that the transition matrix from the basis of quasisymmetric Schur <span><math><mi>Q</mi></math></span>-functions to the basis of peak Young quasisymmetric Schur functions is upper triangular, with entries being non-negative integers. Furthermore, we provide explicit descriptions of the expansion of peak Young quasisymmetric Schur functions in specific cases, in terms of quasisymmetric Schur <span><math><mi>Q</mi></math></span>-functions. We also investigate the combinatorial properties of standard peak immaculate tableaux, standard Young composition tableaux, and standard peak Young composition tableaux. We provide a hook length formula for <span><math><mrow><mi>SPIT</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span> and show that standard Young composition tableaux and standard peak Young composition tableaux can be each bijectively mapped to words satisfying suitable conditions. Especially, cases of compositions with rectangular shape are examined in detail.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104213"},"PeriodicalIF":1.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144571712","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":"Transitive and Gallai colorings of the complete graph","authors":"Ron M. Adin ,&nbsp;Arkady Berenstein ,&nbsp;Jacob Greenstein ,&nbsp;Jian-Rong Li ,&nbsp;Avichai Marmor ,&nbsp;Yuval Roichman","doi":"10.1016/j.ejc.2025.104225","DOIUrl":"10.1016/j.ejc.2025.104225","url":null,"abstract":"<div><div>A Gallai coloring of the complete graph is an edge-coloring with no rainbow triangle. This concept first appeared in the study of incomparability graphs and anti-Ramsey theory. A directed analogue, called transitive coloring, was introduced by Berenstein, Greenstein and Li in a rather general setting. It is studied here for the acyclic tournament. The interplay of the two notions yields new enumerative results and algebraic perspectives.</div><div>We first count Gallai and transitive colorings of the complete graph which use the maximal number of colors. The quasisymmetric generating functions of these colorings, equipped with a natural descent set, are shown to be Schur-positive for any number of colors. Explicit Schur expansions are described when the number of colors is maximal. It follows that descent sets of maximal Gallai and transitive colorings are equidistributed with descent sets of perfect matchings and pattern-avoiding indecomposable permutations, respectively.</div><div>Corresponding commutative algebras are also studied. Their dimensions are shown to be equal to the number of Gallai colorings of the complete graph and the number of transitive colorings of the acyclic tournament, respectively. Relations to Orlik-Terao algebras are established.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104225"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144878330","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":"A consistent sandpile torsor algorithm for regular matroids","authors":"Changxin Ding ,&nbsp;Alex McDonough ,&nbsp;Lilla Tóthmérész ,&nbsp;Chi Ho Yuen","doi":"10.1016/j.ejc.2025.104218","DOIUrl":"10.1016/j.ejc.2025.104218","url":null,"abstract":"<div><div>Every regular matroid is associated with a <em>sandpile group</em>, which acts simply transitively on the set of bases in various ways. Ganguly and the second author introduced the notion of <em>consistency</em> to describe classes of actions that respect deletion–contraction in a precise sense, and proved the consistency of rotor-routing torsors (and uniqueness thereof) for plane graphs.</div><div>In this work, we prove that the class of actions introduced by Backman, Baker, and the fourth author, is consistent for regular matroids. More precisely, we prove the consistency of its generalization given by Backman, Santos and the fourth author, and independently by the first author. This extends the above existence assertion, as well as makes progress on the goal of classifying all consistent actions.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104218"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144831559","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":"Splitter theorems for graph immersions","authors":"Matt DeVos,&nbsp;Mahdieh Malekian","doi":"10.1016/j.ejc.2025.104223","DOIUrl":"10.1016/j.ejc.2025.104223","url":null,"abstract":"<div><div>We establish splitter theorems for graph immersions for two families of graphs, <span><math><mi>k</mi></math></span>-edge-connected graphs, with <span><math><mi>k</mi></math></span> even, and 3-edge-connected, internally 4-edge-connected graphs. As a corollary, we prove that every 3-edge-connected, internally 4-edge-connected graph on at least seven vertices that immerses <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> also has <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>3</mn></mrow></msub></math></span> as an immersion.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104223"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144763810","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":"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":"2025-12-01","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":"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":"2025-12-01","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}