Pub Date : 2026-01-01Epub Date: 2025-09-17DOI: 10.1016/j.ejc.2025.104242
Avichai Cohen, Shaul Zemel
Given a partition of a number , it is known that by adding a long line of length , the dimension of the associated representation of is an integer-valued polynomial of degree in . We show that its expansion in the binomial basis is bounded by the length of , and that the resulting coefficient of index , with alternating signs, counts the standard Young tableaux of shape in which a given collection of consecutive 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 numbers used.
{"title":"Polynomial expressions for the dimensions of the representations of symmetric groups and restricted standard Young tableaux","authors":"Avichai Cohen, 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}
Pub Date : 2025-12-01Epub Date: 2025-08-05DOI: 10.1016/j.ejc.2025.104226
Guantao Chen , Xingyu Lei , Shuchao Li
<div><div>For a given graph <span><math><mi>H</mi></math></span>, we say that a graph <span><math><mi>G</mi></math></span> is <span><math><mi>H</mi></math></span><em>-free</em> if it does not contain <span><math><mi>H</mi></math></span> as a subgraph. Let <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> (resp. <span><math><mrow><msub><mrow><mtext>ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>) denote the maximum size (resp. spectral radius) of an <span><math><mi>n</mi></math></span>-vertex <span><math><mi>H</mi></math></span>-free graph, and <span><math><mrow><mtext>Ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> (resp. <span><math><mrow><msub><mrow><mtext>Ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>) denote the set of all <span><math><mi>n</mi></math></span>-vertex <span><math><mi>H</mi></math></span>-free graphs with <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> edges (resp. spectral radius <span><math><mrow><msub><mrow><mtext>ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>). We call <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> (resp. <span><math><mrow><msub><mrow><mtext>ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>) the <em>Turán number</em> (resp. <em>spectral Turán number</em>) of <span><math><mi>H</mi></math></span>. Suppose that we know the exact values of Turán numbers of <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span>, respectively. Can we get the exact value of the Turán number of the disjoint union of <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><mo>⋯</mo><mo>∪</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span>? Moon considered the disjoint union of complete graphs. A graph <span><math><mi>G</mi></math></span> is <em>color-critical</em> if there exists an edge <span><math><mi>e</mi></math></span> such that <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>−</mo><mi>e</mi><mo>)</mo></mrow><mo><</mo><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Simonovits extended Moon’s result to the disjoint union of <em>color-critical graphs</em> for sufficiently large <span><math><mi>n</mi></math></span>. Erdős et al. determined the Turán number of triangles sha
对于给定的图H，如果图G不包含H作为子图，我们说它是无H的。设ex(n，H) (p。exsp(n，H))表示最大大小。谱半径),Ex(n，H) (resp。Exsp(n，H))表示所有边为ex（n，H）的n顶点无H图的集合。谱半径exsp(n，H))。我们称ex(n，H) (p。exp (n，H)) Turán number （p. 0）假设我们分别知道G1，…，Gk的Turán个数的确切值。我们能否得到G1∪∪Gk的Turán个数的确切值？Moon考虑了完全图的不相交并。如果存在一条边e使得χ(G−e)<χ(G)，则图G是颜色临界的。Simonovits将Moon的结果推广到足够大n的色临界图的不相交并。Erdős等人确定了Turán刚好共享一个顶点的三角形的数量。Chen等人将结果扩展到只共享一个顶点的完全图。设F是Fi的一个不相交并，其中Fi是将一个顶点与所有的顶点联结在一起得到的一个图，每个Fij都是一个色临界图。对于较大的n，我们确定了ex（n，F）和exsp（n，F）。此外，我们证明了对于这些图中的每一个F，只要n足够大，Exsp（n，F）任任（n，F），这就提供了一大类图，对Liu和Ning最近提出的一个开放问题给出了一个正答案：刻画满足Exsp（n，F）任任（n，F）的图F。
{"title":"The exact Turán number of disjoint graphs– A generalization of Simonovits’ theorem, and beyond","authors":"Guantao Chen , Xingyu Lei , Shuchao Li","doi":"10.1016/j.ejc.2025.104226","DOIUrl":"10.1016/j.ejc.2025.104226","url":null,"abstract":"<div><div>For a given graph <span><math><mi>H</mi></math></span>, we say that a graph <span><math><mi>G</mi></math></span> is <span><math><mi>H</mi></math></span><em>-free</em> if it does not contain <span><math><mi>H</mi></math></span> as a subgraph. Let <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> (resp. <span><math><mrow><msub><mrow><mtext>ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>) denote the maximum size (resp. spectral radius) of an <span><math><mi>n</mi></math></span>-vertex <span><math><mi>H</mi></math></span>-free graph, and <span><math><mrow><mtext>Ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> (resp. <span><math><mrow><msub><mrow><mtext>Ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>) denote the set of all <span><math><mi>n</mi></math></span>-vertex <span><math><mi>H</mi></math></span>-free graphs with <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> edges (resp. spectral radius <span><math><mrow><msub><mrow><mtext>ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>). We call <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> (resp. <span><math><mrow><msub><mrow><mtext>ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>) the <em>Turán number</em> (resp. <em>spectral Turán number</em>) of <span><math><mi>H</mi></math></span>. Suppose that we know the exact values of Turán numbers of <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span>, respectively. Can we get the exact value of the Turán number of the disjoint union of <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><mo>⋯</mo><mo>∪</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span>? Moon considered the disjoint union of complete graphs. A graph <span><math><mi>G</mi></math></span> is <em>color-critical</em> if there exists an edge <span><math><mi>e</mi></math></span> such that <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>−</mo><mi>e</mi><mo>)</mo></mrow><mo><</mo><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Simonovits extended Moon’s result to the disjoint union of <em>color-critical graphs</em> for sufficiently large <span><math><mi>n</mi></math></span>. 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}
Pub Date : 2025-12-01Epub Date: 2025-07-12DOI: 10.1016/j.ejc.2025.104220
Robert Angarone , Daniel Soskin
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.
我们描述了正半定矩阵中广义对角线中的所有不等式。这些结果是由对称群上的一个简单偏序控制的。给出了Drake， Gerrish， and Skandera关于完全非负矩阵中广义对角线间不等式的一个类似结果。
{"title":"Generalized diagonals in positive semi-definite matrices","authors":"Robert Angarone , 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}
Pub Date : 2025-12-01Epub Date: 2025-06-09DOI: 10.1016/j.ejc.2025.104190
Daniel W. Cranston
Fix a planar graph and a list assignment with for all . Let and be -colorings of . A recoloring sequence from to is a sequence of -colorings, beginning with and ending with , such that each successive pair in the sequence differs in the color on a single vertex of . We show that there exists a constant such that for all choices of and there exists a recoloring sequence from to that recolors each vertex at most times. In particular, has length at most . 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.
{"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}
Pub Date : 2025-12-01Epub Date: 2025-07-08DOI: 10.1016/j.ejc.2025.104213
Seung-Il Choi , Sun-Young Nam , Young-Tak Oh
In this paper, we explore the relationship between quasisymmetric Schur -functions and peak Young quasisymmetric Schur functions. We introduce a bijection on such that and share identical descent distributions. Here, is the set of standard peak immaculate tableaux of shape , and and 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 -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 -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 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.
{"title":"Quasisymmetric Schur Q-functions and peak Young quasisymmetric Schur functions","authors":"Seung-Il Choi , Sun-Young Nam , 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}
Pub Date : 2025-12-01Epub Date: 2025-08-21DOI: 10.1016/j.ejc.2025.104225
Ron M. Adin , Arkady Berenstein , Jacob Greenstein , Jian-Rong Li , Avichai Marmor , Yuval Roichman
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.
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.
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.
{"title":"Transitive and Gallai colorings of the complete graph","authors":"Ron M. Adin , Arkady Berenstein , Jacob Greenstein , Jian-Rong Li , Avichai Marmor , 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}
Pub Date : 2025-12-01Epub Date: 2025-07-16DOI: 10.1016/j.ejc.2025.104218
Changxin Ding , Alex McDonough , Lilla Tóthmérész , Chi Ho Yuen
Every regular matroid is associated with a sandpile group, which acts simply transitively on the set of bases in various ways. Ganguly and the second author introduced the notion of consistency 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.
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.
{"title":"A consistent sandpile torsor algorithm for regular matroids","authors":"Changxin Ding , Alex McDonough , Lilla Tóthmérész , 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}
Pub Date : 2025-12-01Epub Date: 2025-07-08DOI: 10.1016/j.ejc.2025.104214
Sebastian Mies , Benjamin Moore , Evelyne Smith-Roberge
The pseudoforest version of the Strong Nine Dragon Tree Conjecture states that if a graph has maximum average degree at most , then it has a decomposition into pseudoforests where in one pseudoforest the components of have at most 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 is acyclic, the diameter of the components of is at most , where , and at most if . Furthermore, for any component of and any , we have if . 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 to have any constant maximum degree, instead of enforcing every component of to have at most edges.
{"title":"Beyond the pseudoforest strong Nine Dragon Tree Theorem","authors":"Sebastian Mies , Benjamin Moore , 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}
Pub Date : 2025-12-01Epub Date: 2025-08-02DOI: 10.1016/j.ejc.2025.104223
Matt DeVos, Mahdieh Malekian
We establish splitter theorems for graph immersions for two families of graphs, -edge-connected graphs, with 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 also has as an immersion.
{"title":"Splitter theorems for graph immersions","authors":"Matt DeVos, 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}
Pub Date : 2025-12-01Epub Date: 2025-06-18DOI: 10.1016/j.ejc.2025.104200
Chunyang Hu, Shiping Liu
For a non-bipartite finite Cayley graph, we show the non-trivial eigenvalues of its normalized adjacency matrix lie in the interval for some absolute constants and , where 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.
{"title":"Vertex isoperimetry on signed graphs and spectra of non-bipartite Cayley and Cayley sum graphs","authors":"Chunyang Hu, 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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号