{"title":"When (signless) Laplacian coefficients meet matchings of subdivision","authors":"Zhibin Du","doi":"10.1016/j.ejc.2024.104087","DOIUrl":"10.1016/j.ejc.2024.104087","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph, whose subdivision is denoted by <span><math><mrow><mi>S</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Let <span><math><mrow><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>L</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> be the characteristic polynomial of the Laplacian matrix of <span><math><mi>G</mi></math></span>. In 1974, Kelmans and Chelnokov (1974) gave a graph theoretical interpretation for the coefficients of <span><math><mrow><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>L</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>, in terms of the spanning forests of <span><math><mi>G</mi></math></span>. In this paper, we present another graph theoretical interpretation of the Laplacian coefficients by using the matching numbers of <span><math><mrow><mi>S</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, generalizing the cases of trees and unicyclic graphs, which were established by Zhou and Gutman (2008) and Chen and Yan (2021), respectively. Analogously, a graph theoretical interpretation of the signless Laplacian coefficients is also presented, whose previous graph theoretical interpretation is based on the so-called TU-subgraphs (the spanning subgraphs whose components are trees or odd-unicyclic graphs) due to Cvetković et al. (2007). Some formulas related to the number of spanning trees are also given.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104087"},"PeriodicalIF":1.0,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659340","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":"Freehedra are short","authors":"Daria Poliakova","doi":"10.1016/j.ejc.2024.104084","DOIUrl":"10.1016/j.ejc.2024.104084","url":null,"abstract":"<div><div>We prove the combinatorial property of shortness for freehedra. Note that associahedra, a related family of polytopes, are not short.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104084"},"PeriodicalIF":1.0,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659339","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}
{"title":"On the Erdős–Tuza–Valtr conjecture","authors":"Jineon Baek","doi":"10.1016/j.ejc.2024.104085","DOIUrl":"10.1016/j.ejc.2024.104085","url":null,"abstract":"<div><div>The Erdős–Szekeres conjecture states that any set of more than <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup></math></span> points in the plane with no three on a line contains the vertices of a convex <span><math><mi>n</mi></math></span>-gon. Erdős, Tuza, and Valtr strengthened the conjecture by stating that any set of more than <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mi>n</mi><mo>−</mo><mi>b</mi></mrow><mrow><mi>a</mi><mo>−</mo><mn>2</mn></mrow></msubsup><mfenced><mrow><mfrac><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mrow><mi>i</mi></mrow></mfrac></mrow></mfenced></mrow></math></span> points in a plane either contains the vertices of a convex <span><math><mi>n</mi></math></span>-gon, <span><math><mi>a</mi></math></span> points lying on a concave downward curve, or <span><math><mi>b</mi></math></span> points lying on a concave upward curve. They also showed that the generalization is actually equivalent to the Erdős–Szekeres conjecture. We prove the first new case of the Erdős–Tuza–Valtr conjecture since the original 1935 paper of Erdős and Szekeres. Namely, we show that any set of <span><math><mrow><mfenced><mrow><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo>+</mo><mn>2</mn></mrow></math></span> points in the plane with no three points on a line and no two points sharing the same <span><math><mi>x</mi></math></span>-coordinate either contains 4 points lying on a concave downward curve or the vertices of a convex <span><math><mi>n</mi></math></span>-gon. The proof is also formalized in <em>Lean 4</em>, a computer proof assistance, to ensure the correctness of the proof.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104085"},"PeriodicalIF":1.0,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659338","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 combinatorial PROP for bialgebras","authors":"Jorge Becerra","doi":"10.1016/j.ejc.2024.104086","DOIUrl":"10.1016/j.ejc.2024.104086","url":null,"abstract":"<div><div>It is a classical result that the category of finitely-generated free monoids serves as a PROP for commutative bialgebras. Attaching permutations to fix the order of multiplication, we construct an extension of this category that is equivalent to the PROP for bialgebras.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104086"},"PeriodicalIF":1.0,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586709","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}
{"title":"Signed Mahonian polynomials on derangements in classical Weyl groups","authors":"Kathy Q. Ji , Dax T.X. Zhang","doi":"10.1016/j.ejc.2024.104083","DOIUrl":"10.1016/j.ejc.2024.104083","url":null,"abstract":"<div><div>The polynomial of the major index <span><math><mrow><msub><mrow><mi>maj</mi></mrow><mrow><mi>W</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></mrow></math></span> over the subset <span><math><mi>T</mi></math></span> of the Coxeter group <span><math><mi>W</mi></math></span> is called the Mahonian polynomial over <span><math><mi>T</mi></math></span>, where <span><math><mrow><msub><mrow><mi>maj</mi></mrow><mrow><mi>W</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></mrow></math></span> is a Mahonian statistic of an element <span><math><mrow><mi>σ</mi><mo>∈</mo><mi>T</mi></mrow></math></span>, whereas the polynomial of the major index <span><math><mrow><msub><mrow><mi>maj</mi></mrow><mrow><mi>W</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></mrow></math></span> with the sign <span><math><msup><mrow><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>W</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></mrow></msup></math></span> over the subset <span><math><mi>T</mi></math></span> is referred to as the signed Mahonian polynomial over <span><math><mi>T</mi></math></span>, where <span><math><mrow><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>W</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></mrow></math></span> is the length of <span><math><mrow><mi>σ</mi><mo>∈</mo><mi>T</mi></mrow></math></span>. Gessel, Wachs, and Chow established formulas for the Mahonian polynomials over the sets of derangements in the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and the hyperoctahedral group <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. By extending Wachs’ approach and employing a refinement of Stanley’s shuffle theorem established in our recent paper (Ji and Zhang, 2024), we derive a formula for the Mahonian polynomials over the set of derangements in the even-signed permutation group <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. This completes a picture which is now known for all the classical Weyl groups. Gessel–Simion, Adin–Gessel–Roichman, and Biagioli previously established formulas for the signed Mahonian polynomials over the classical Weyl groups. Building upon their formulas, we derive some new formulas for the signed Mahonian polynomials over the set of derangements in classical Weyl groups. As applications of the formulas for the (signed) Mahonian polynomials over the sets of derangements in the classical Weyl groups, we obtain enumerative formulas of the number of derangements in classical Weyl groups with even lengths.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104083"},"PeriodicalIF":1.0,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142577928","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}
Lucas Aragão , João Pedro Marciano , Walner Mendonça
{"title":"Degree conditions for Ramsey goodness of paths","authors":"Lucas Aragão , João Pedro Marciano , Walner Mendonça","doi":"10.1016/j.ejc.2024.104082","DOIUrl":"10.1016/j.ejc.2024.104082","url":null,"abstract":"<div><div>A classical result of Chvátal implies that if <span><math><mrow><mi>n</mi><mo>≥</mo><mrow><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span>, then any colouring of the edges of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> in red and blue contains either a monochromatic red <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> or a monochromatic blue <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>. We study a natural generalisation of his result, determining the exact minimum degree condition for a graph <span><math><mi>G</mi></math></span> on <span><math><mrow><mi>n</mi><mo>=</mo><mrow><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span> vertices which guarantees that the same Ramsey property holds in <span><math><mi>G</mi></math></span>. In particular, using a slight generalisation of a result of Haxell, we show that <span><math><mrow><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mi>n</mi><mo>−</mo><mfenced><mrow><mi>t</mi><mo>/</mo><mn>2</mn></mrow></mfenced></mrow></math></span> suffices, and that this bound is best possible. We also use a classical result of Bollobás, Erdős, and Straus to prove a tight minimum degree condition in the case <span><math><mrow><mi>r</mi><mo>=</mo><mn>3</mn></mrow></math></span> for all <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn><mi>t</mi><mo>−</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104082"},"PeriodicalIF":1.0,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533813","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 faces of unigraphic 3-polytopes","authors":"Riccardo W. Maffucci","doi":"10.1016/j.ejc.2024.104081","DOIUrl":"10.1016/j.ejc.2024.104081","url":null,"abstract":"<div><div>A 3-polytope is a 3-connected, planar graph. It is called unigraphic if it does not share its vertex degree sequence with any other 3-polytope, up to graph isomorphism. The classification of unigraphic 3-polytopes appears to be a difficult problem.</div><div>In this paper we prove that, apart from pyramids, all unigraphic 3-polytopes have no <span><math><mi>n</mi></math></span>-gonal faces for <span><math><mrow><mi>n</mi><mo>≥</mo><mn>10</mn></mrow></math></span>. Our method involves defining several planar graph transformations on a given 3-polytope containing an <span><math><mi>n</mi></math></span>-gonal face with <span><math><mrow><mi>n</mi><mo>≥</mo><mn>10</mn></mrow></math></span>. The delicate part is to prove that, for every such 3-polytope, at least one of these transformations both preserves 3-connectivity, and is not an isomorphism.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104081"},"PeriodicalIF":1.0,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142441010","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":"Bounded unique representation bases for the integers","authors":"Yong-Gao Chen, Jin-Hui Fang","doi":"10.1016/j.ejc.2024.104080","DOIUrl":"10.1016/j.ejc.2024.104080","url":null,"abstract":"<div><div>For a nonempty set <span><math><mi>A</mi></math></span> of integers and an integer <span><math><mi>n</mi></math></span>, let <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> be the number of representations of <span><math><mrow><mi>n</mi><mo>=</mo><mi>a</mi><mo>+</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> with <span><math><mrow><mi>a</mi><mo>≤</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> and <span><math><mrow><mi>a</mi><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>A</mi></mrow></math></span>, and let <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> be the number of representations of <span><math><mrow><mi>n</mi><mo>=</mo><mi>a</mi><mo>−</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> with <span><math><mrow><mi>a</mi><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>A</mi></mrow></math></span>. Erdős and Turán (1941) posed the profound conjecture: if <span><math><mi>A</mi></math></span> is a set of positive integers such that <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>≥</mo><mn>1</mn></mrow></math></span> for all sufficiently large <span><math><mi>n</mi></math></span>, then <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> is unbounded. Nešetřil and Serra (2004) introduced the notion of bounded sets and confirmed the Erdős–Turán conjecture for all bounded bases. Nathanson (2003) considered the existence of the set <span><math><mi>A</mi></math></span> with logarithmic growth such that <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> for all integers <span><math><mi>n</mi></math></span>. In this paper, we prove that, for any positive function <span><math><mrow><mi>l</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>l</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>→</mo><mn>0</mn></mrow></math></span> as <span><math><mrow><mi>x</mi><mo>→</mo><mi>∞</mi></mrow></math></span>, there is a bounded set <span><math><mi>A</mi></math></span> of integers such that <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> for all integers <span><math><mi>n</mi></math></span> and <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> for all positi","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104080"},"PeriodicalIF":1.0,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142441009","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}
Jacob Fox , Tung Nguyen , Alex Scott , Paul Seymour
{"title":"Induced subgraph density. II. Sparse and dense sets in cographs","authors":"Jacob Fox , Tung Nguyen , Alex Scott , Paul Seymour","doi":"10.1016/j.ejc.2024.104075","DOIUrl":"10.1016/j.ejc.2024.104075","url":null,"abstract":"<div><div>A well-known theorem of Rödl says that for every graph <span><math><mi>H</mi></math></span>, and every <span><math><mrow><mi>ɛ</mi><mo>></mo><mn>0</mn></mrow></math></span>, there exists <span><math><mrow><mi>δ</mi><mo>></mo><mn>0</mn></mrow></math></span> such that if <span><math><mi>G</mi></math></span> does not contain an induced copy of <span><math><mi>H</mi></math></span>, then there exists <span><math><mrow><mi>X</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>≥</mo><mi>δ</mi><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow></mrow></math></span> such that one of <span><math><mrow><mi>G</mi><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow><mo>,</mo><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow></mrow></math></span> has edge-density at most <span><math><mi>ɛ</mi></math></span>. But how does <span><math><mi>δ</mi></math></span> depend on <span><math><mi>ϵ</mi></math></span>? Fox and Sudakov conjectured that the dependence is at most polynomial: that for all <span><math><mi>H</mi></math></span> there exists <span><math><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></math></span> such that for all <span><math><mi>ɛ</mi></math></span> with <span><math><mrow><mn>0</mn><mo><</mo><mi>ɛ</mi><mo>≤</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math></span>, Rödl’s theorem holds with <span><math><mrow><mi>δ</mi><mo>=</mo><msup><mrow><mi>ɛ</mi></mrow><mrow><mi>c</mi></mrow></msup></mrow></math></span>. This conjecture implies the Erdős–Hajnal conjecture, and until now it had not been verified for any non-trivial graphs <span><math><mi>H</mi></math></span>. Our first result shows that it is true when <span><math><mrow><mi>H</mi><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></math></span>. Indeed, in that case we can take <span><math><mrow><mi>δ</mi><mo>=</mo><mi>ɛ</mi></mrow></math></span>, and insist that one of <span><math><mrow><mi>G</mi><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow><mo>,</mo><mover><mrow><mi>G</mi></mrow><mo>¯</mo></mover><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow></mrow></math></span> has maximum degree at most <span><math><mrow><msup><mrow><mi>ɛ</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow></mrow></math></span>).</div><div>Second, we will show that every graph <span><math><mi>H</mi></math></span> that can be obtained by substitution from copies of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> satisfies the Fox–Sudakov conjecture. To prove this, we need to work with a stronger property. Let us say <span><math><mi>H</mi></math></span> is <em>viral</em> if there exists <span><math><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></math></span> such that for all <span><math><mi>ɛ</mi></math></span> with <span><math><mrow><mn>0</mn><mo><</mo><mi>ɛ</mi><mo>≤</mo><mn>1</mn><mo>/","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104075"},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142423576","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}
Lucas Aragão , Maurício Collares , Gabriel Dahia , João Pedro Marciano
{"title":"The diameter of randomly twisted hypercubes","authors":"Lucas Aragão , Maurício Collares , Gabriel Dahia , João Pedro Marciano","doi":"10.1016/j.ejc.2024.104078","DOIUrl":"10.1016/j.ejc.2024.104078","url":null,"abstract":"<div><div>The <span><math><mi>n</mi></math></span>-dimensional random twisted hypercube <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is constructed recursively by taking two instances of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>, with any joint distribution, and adding a random perfect matching between their vertex sets. Benjamini, Dikstein, Gross, and Zhukovskii showed that its diameter is <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mo>log</mo><mo>log</mo><mo>log</mo><mi>n</mi><mo>/</mo><mo>log</mo><mo>log</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> with high probability and at least <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi></mrow></math></span>. We improve their upper bound by showing that <span><math><mrow><mi>diam</mi><mrow><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mfrac><mrow><mi>n</mi></mrow><mrow><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi></mrow></mfrac></mrow></math></span> with high probability.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104078"},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142423587","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}
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