European Journal of Combinatorics最新文献

{"title":"Spectral supersaturation: Triangles and bowties","authors":"Yongtao Li , Lihua Feng , Yuejian Peng","doi":"10.1016/j.ejc.2025.104171","DOIUrl":"10.1016/j.ejc.2025.104171","url":null,"abstract":"<div><div>A classical result of Erdős and Rademacher (1955) demonstrates a fundamental supersaturation phenomenon in extremal combinatorics: every graph on <span><math><mi>n</mi></math></span> vertices with more than <span><math><mrow><mo>⌊</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>4</mn><mo>⌋</mo></mrow></math></span> edges contains at least <span><math><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow></math></span> triangles. Let <span><math><mrow><mi>λ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the spectral radius of the adjacency matrix of a graph <span><math><mi>G</mi></math></span>. Recently, Ning and Zhai (2023) proved that every <span><math><mi>n</mi></math></span>-vertex graph <span><math><mi>G</mi></math></span> with <span><math><mrow><mi>λ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><msqrt><mrow><mrow><mo>⌊</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>4</mn><mo>⌋</mo></mrow></mrow></msqrt></mrow></math></span> contains at least <span><math><mrow><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span> triangles, unless <span><math><mi>G</mi></math></span> is a balanced complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mo>,</mo><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow></mrow></msub></math></span>. The aim of this paper is two-fold. Using a different approach which we term the supersaturation-stability method, we prove a stability variant of the Ning–Zhai result by showing that such a graph <span><math><mi>G</mi></math></span> contains at least <span><math><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></math></span> triangles if no vertex lies in all triangles of <span><math><mi>G</mi></math></span>. This bound is the best possible and it could also be viewed as a spectral analogue of a theorem of Xiao and Katona (2021), which guarantees <span><math><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></math></span> triangles under the assumption that <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>></mo><mrow><mo>⌊</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>4</mn><mo>⌋</mo></mrow></mrow></math></span> and no vertex is in all triangles of <span><math><mi>G</mi></math></span>.</div><div>The second part concerns with the spectral supersaturation for the bowtie, which consists of two triangles sharing a vertex. Erdős, Füredi, Gould and Gunderson (1995) proved that every <span><math><mi>n</mi></math></span>-vertex graph with more than <span><math><mrow><mrow><mo>⌊</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>4</mn><mo>⌋</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span> edges contains a bowtie. The spectral supersaturation ","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"128 ","pages":"Article 104171"},"PeriodicalIF":1.0,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143912503","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":"Number of facets of symmetric edge polytopes arising from join graphs","authors":"Aki Mori ,&nbsp;Kenta Mori ,&nbsp;Hidefumi Ohsugi","doi":"10.1016/j.ejc.2025.104165","DOIUrl":"10.1016/j.ejc.2025.104165","url":null,"abstract":"<div><div>Symmetric edge polytopes of graphs are important object in Ehrhart theory, and have an application to Kuramoto models. In the present paper, we study the upper and lower bounds for the number of facets of symmetric edge polytopes of connected graphs conjectured by Braun and Bruegge. In particular, we show that their conjecture is true for any graph that is the join of two graphs (equivalently, for any connected graph whose complement graph is not connected). It is known that any symmetric edge polytope is a centrally symmetric reflexive polytope. Hence our results give a partial answer to Nill’s conjecture: the number of facets of a <span><math><mi>d</mi></math></span>-dimensional reflexive polytope is at most <span><math><msup><mrow><mn>6</mn></mrow><mrow><mi>d</mi><mo>/</mo><mn>2</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104165"},"PeriodicalIF":1.0,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143904194","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 Δ-edge stability number of graphs","authors":"Saieed Akbari ,&nbsp;Reza Hosseini Dolatabadi ,&nbsp;Mohsen Jamaali ,&nbsp;Sandi Klavžar ,&nbsp;Nazanin Movarraei","doi":"10.1016/j.ejc.2025.104167","DOIUrl":"10.1016/j.ejc.2025.104167","url":null,"abstract":"<div><div>The <span><math><mi>Δ</mi></math></span>-edge stability number <span><math><mrow><msub><mrow><mi>es</mi></mrow><mrow><mi>Δ</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of a graph <span><math><mi>G</mi></math></span> is the minimum number of edges of <span><math><mi>G</mi></math></span> whose removal results in a subgraph <span><math><mi>H</mi></math></span> with <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>=</mo><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span>. Sets whose removal results in a subgraph with smaller maximum degree are called mitigating sets. It is proved that there always exists a mitigating set which induces a disjoint union of paths of order 2 or 3. Minimum mitigating sets which induce matchings are characterized. It is proved that to obtain an upper bound of the form <span><math><mrow><msub><mrow><mi>es</mi></mrow><mrow><mi>Δ</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>c</mi><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span> for an arbitrary graph <span><math><mi>G</mi></math></span> of given maximum degree <span><math><mi>Δ</mi></math></span>, where <span><math><mi>c</mi></math></span> is a given constant, it suffices to prove the bound for <span><math><mi>Δ</mi></math></span>-regular graphs. Sharp upper bounds of this form are derived for regular graphs. It is proved that if <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mfrac><mrow><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>−</mo><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></math></span> or the induced subgraph on maximum degree vertices has a <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>-edge coloring, then <span><math><mrow><msub><mrow><mi>es</mi></mrow><mrow><mi>Δ</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mrow><mo>⌈</mo><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>/</mo><mn>2</mn><mo>⌉</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104167"},"PeriodicalIF":1.0,"publicationDate":"2025-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143898757","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":"Comparing width parameters on graph classes","authors":"Nick Brettell ,&nbsp;Andrea Munaro ,&nbsp;Daniël Paulusma ,&nbsp;Shizhou Yang","doi":"10.1016/j.ejc.2025.104163","DOIUrl":"10.1016/j.ejc.2025.104163","url":null,"abstract":"<div><div>We study how the relationship between non-equivalent width parameters changes once we restrict to some special graph class. As width parameters we consider treewidth, clique-width, twin-width, mim-width, sim-width and tree-independence number, whereas as graph classes we consider <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span>-subgraph-free graphs, line graphs and their common superclass, for <span><math><mrow><mi>t</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span>-free graphs. For <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span>-subgraph-free graphs, we extend a known result of Gurski and Wanke (2000) and provide a complete comparison, showing in particular that treewidth, clique-width, mim-width, sim-width and tree-independence number are all equivalent. For line graphs, we extend a result of Gurski and Wanke (2007) and also provide a complete comparison, showing in particular that clique-width, mim-width, sim-width and tree-independence number are all equivalent, and bounded if and only if the class of root graphs has bounded treewidth. For <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span>-free graphs, we provide an almost-complete comparison, leaving open only one missing case. We show in particular that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span>-free graphs of bounded mim-width have bounded tree-independence number, and obtain structural and algorithmic consequences of this result, such as a proof of a special case of a recent conjecture of Dallard, Milanič and Štorgel. Finally, we consider the question of whether boundedness of a certain width parameter is preserved under graph powers. We show that this question has a positive answer for sim-width precisely in the case of odd powers.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104163"},"PeriodicalIF":1.0,"publicationDate":"2025-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143898756","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 sl(2)-weight system at c = 3/8 for graphs","authors":"Daniil Fomichev ,&nbsp;Maksim Karev","doi":"10.1016/j.ejc.2025.104160","DOIUrl":"10.1016/j.ejc.2025.104160","url":null,"abstract":"<div><div>We construct a 4-invariant that extends the specialization of the <span><math><mrow><mi>sl</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>-weight system at <span><math><mrow><mi>c</mi><mo>=</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>8</mn></mrow></mfrac></mrow></math></span> and satisfies a simple deletion–contraction relation.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104160"},"PeriodicalIF":1.0,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143895975","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":"Posets, their incidence algebras and relative operads, and the cohomology comparison theorem","authors":"V. Jacky III Batkam Mbatchou ,&nbsp;Frédéric Patras ,&nbsp;Calvin Tcheka","doi":"10.1016/j.ejc.2025.104162","DOIUrl":"10.1016/j.ejc.2025.104162","url":null,"abstract":"<div><div>Motivated by various developments in algebraic combinatorics and its applications, we investigate here the fine structure of a fundamental but little known theorem, the Gerstenhaber and Schack cohomology comparison theorem. The theorem classically asserts that there is a cochain equivalence between the usual singular cochain complex of a simplicial complex and the relative Hochschild complex of its incidence algebra, and a quasi-isomorphism with the standard Hochschild complex. Here, we will be mostly interested in its application to arbitrary posets (or, equivalently, finite topological spaces satisfying the <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> separation axiom) and their incidence algebras. We construct various structures, classical and new, on the above two complexes: cosimplicial, differential graded algebra, operadic and brace algebra structures and show that the comparison theorem preserves all of them. These results provide non standard insights on links between the theory of posets, incidence algebras, endomorphism operads and finite and combinatorial topology. By <em>non standard</em>, we refer here to the use of <em>relative</em> versions of Hochschild complexes and operads.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104162"},"PeriodicalIF":1.0,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143885987","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 diameter of random Schreier graphs","authors":"Daniele Dona ,&nbsp;Luca Sabatini","doi":"10.1016/j.ejc.2025.104164","DOIUrl":"10.1016/j.ejc.2025.104164","url":null,"abstract":"<div><div>We give a combinatorial proof of the following theorem. Let <span><math><mi>G</mi></math></span> be any finite group acting transitively on a set of cardinality <span><math><mi>n</mi></math></span>. If <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>G</mi></mrow></math></span> is a random set of size <span><math><mi>k</mi></math></span>, with <span><math><mrow><mi>k</mi><mo>≥</mo><msup><mrow><mrow><mo>(</mo><mo>log</mo><mi>n</mi><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>+</mo><mi>ɛ</mi></mrow></msup></mrow></math></span> for some <span><math><mrow><mi>ɛ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>, then the diameter of the corresponding Schreier graph is <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msub><mrow><mo>log</mo></mrow><mrow><mi>k</mi></mrow></msub><mi>n</mi><mo>)</mo></mrow></mrow></math></span> with high probability. Except for the implicit constant, this result is the best possible.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104164"},"PeriodicalIF":1.0,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143885988","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":"Refined canonical stable Grothendieck polynomials and their duals, Part 2","authors":"Byung-Hak Hwang ,&nbsp;Jihyeug Jang ,&nbsp;Jang Soo Kim ,&nbsp;Minho Song ,&nbsp;U-Keun Song","doi":"10.1016/j.ejc.2025.104166","DOIUrl":"10.1016/j.ejc.2025.104166","url":null,"abstract":"<div><div>This paper is the sequel of the paper under the same title with part 1, where we introduced refined canonical stable Grothendieck polynomials and their duals with two families of infinite parameters. In this paper we give combinatorial interpretations for these polynomials using generalizations of set-valued tableaux and reverse plane partitions, respectively. Our results extend to their flagged and skew versions.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104166"},"PeriodicalIF":1.0,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143885982","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":"When all directed cycles have length three","authors":"Paul Seymour","doi":"10.1016/j.ejc.2025.104161","DOIUrl":"10.1016/j.ejc.2025.104161","url":null,"abstract":"<div><div>We give a construction to build all digraphs with the property that every directed cycle has length three.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104161"},"PeriodicalIF":1.0,"publicationDate":"2025-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143873957","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":"Grothendieck polynomials of inverse fireworks permutations","authors":"Chen-An (Jack) Chou ,&nbsp;Tianyi Yu","doi":"10.1016/j.ejc.2025.104158","DOIUrl":"10.1016/j.ejc.2025.104158","url":null,"abstract":"<div><div>Pipe dreams are combinatorial objects that compute Grothendieck polynomials. We introduce a new combinatorial object that naturally recasts the pipe dream formula. From this, we obtain the first direct combinatorial formula for the top degree components of Grothendieck polynomials, also known as the Castelnuovo–Mumford polynomials. We also prove the inverse fireworks case of a conjecture of Mészáros, Setiabrata, and St. Dizier on the support of Grothendieck polynomials.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104158"},"PeriodicalIF":1.0,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143784054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}