{"title":"On the maximum number of common neighbours in dense random regular graphs","authors":"Mikhail Isaev , Maksim Zhukovskii","doi":"10.1016/j.ejc.2024.104106","DOIUrl":"10.1016/j.ejc.2024.104106","url":null,"abstract":"<div><div>We derive the distribution of the maximum number of common neighbours of a pair of vertices in a dense random regular graph. The proof involves two important steps. One step is to establish the extremal independence property: the asymptotic equivalence with the maximum component of a vector with independent marginal distributions. The other step is to prove that the distribution of the number of common neighbours for each pair of vertices can be approximated by the binomial distribution.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104106"},"PeriodicalIF":1.0,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176365","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}
Alvaro Carbonero , Hidde Koerts , Benjamin Moore , Sophie Spirkl
{"title":"On heroes in digraphs with forbidden induced forests","authors":"Alvaro Carbonero , Hidde Koerts , Benjamin Moore , Sophie Spirkl","doi":"10.1016/j.ejc.2024.104104","DOIUrl":"10.1016/j.ejc.2024.104104","url":null,"abstract":"<div><div>We continue a line of research which studies which hereditary families of digraphs have bounded dichromatic number. For a class of digraphs <span><math><mi>C</mi></math></span>, a hero in <span><math><mi>C</mi></math></span> is any digraph <span><math><mi>H</mi></math></span> such that <span><math><mi>H</mi></math></span>-free digraphs in <span><math><mi>C</mi></math></span> have bounded dichromatic number. We show that if <span><math><mi>F</mi></math></span> is an oriented star of degree at least five, the only heroes for the class of <span><math><mi>F</mi></math></span>-free digraphs are transitive tournaments. For oriented stars <span><math><mi>F</mi></math></span> of degree exactly four, we show the only heroes in <span><math><mi>F</mi></math></span>-free digraphs are transitive tournaments, or possibly special joins of transitive tournaments. Aboulker et al. characterized the set of heroes of <span><math><mrow><mo>{</mo><mi>H</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mover><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mo>→</mo></mover><mo>}</mo></mrow></math></span>-free digraphs almost completely, and we show the same characterization for the class of <span><math><mrow><mo>{</mo><mi>H</mi><mo>,</mo><mi>r</mi><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mover><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mo>→</mo></mover><mo>}</mo></mrow></math></span>-free digraphs. Lastly, we show that if we forbid two “valid” orientations of brooms, then every transitive tournament is a hero for this class of digraphs.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"125 ","pages":"Article 104104"},"PeriodicalIF":1.0,"publicationDate":"2024-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136075","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":"A group action on cyclic compositions and γ-positivity","authors":"Shishuo Fu , Jie Yang","doi":"10.1016/j.ejc.2024.104107","DOIUrl":"10.1016/j.ejc.2024.104107","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>m</mi></mrow></msub></math></span> be the number of Dyck paths of semilength <span><math><mi>n</mi></math></span> with <span><math><mi>k</mi></math></span> occurrences of <span><math><mrow><mi>U</mi><mi>D</mi></mrow></math></span> and <span><math><mi>m</mi></math></span> occurrences of <span><math><mrow><mi>U</mi><mi>U</mi><mi>D</mi></mrow></math></span>. We establish in two ways a new interpretation of the numbers <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>m</mi></mrow></msub></math></span> in terms of plane trees and internal nodes. The first way builds on a new characterization of plane trees that involves cyclic compositions. The second proof utilizes a known interpretation of <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>m</mi></mrow></msub></math></span> in terms of plane trees and leaves, and a recent involution on plane trees constructed by Li, Lin, and Zhao. Moreover, a group action on the set of cyclic compositions (or equivalently, 2-dominant compositions) is introduced, which amounts to give a combinatorial proof of the <span><math><mi>γ</mi></math></span>-positivity of the Narayana polynomial, as well as the <span><math><mi>γ</mi></math></span>-positivity of the polynomial <span><math><mrow><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>≔</mo><msub><mrow><mo>∑</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mi>k</mi></mrow></msub><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>,</mo><mi>m</mi></mrow></msub><msup><mrow><mi>t</mi></mrow><mrow><mi>m</mi></mrow></msup></mrow></math></span> previously obtained by Bóna et al, with apparently new combinatorial interpretations of their <span><math><mi>γ</mi></math></span>-coefficients.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"125 ","pages":"Article 104107"},"PeriodicalIF":1.0,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136446","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":"Coloring minimal Cayley graphs","authors":"Ignacio García-Marco , Kolja Knauer","doi":"10.1016/j.ejc.2024.104108","DOIUrl":"10.1016/j.ejc.2024.104108","url":null,"abstract":"<div><div>In 1978 Babai raised the question whether all minimal Cayley graphs have bounded chromatic number; in 1994 he conjectured a negative answer. In this paper we show that any minimal Cayley graph of a (finitely generated) generalized dihedral or nilpotent group has chromatic number at most 3, while 4 colors are sometimes necessary for soluble groups. On the other hand we address a related question proposed by Babai in 1978 by constructing graphs of unbounded chromatic number that admit a proper edge coloring such that each cycle has some color at least twice. The latter can be viewed as a step towards confirming Babai’s 1994 conjecture – a problem that remains open.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"125 ","pages":"Article 104108"},"PeriodicalIF":1.0,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136445","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":"Progressions in Euclidean Ramsey theory","authors":"Jakob Führer , Géza Tóth","doi":"10.1016/j.ejc.2024.104105","DOIUrl":"10.1016/j.ejc.2024.104105","url":null,"abstract":"<div><div>Conlon and Wu (2022) showed that there is a red/blue-coloring of <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> that does not contain 3 red collinear points separated by unit distance and <span><math><mrow><mi>m</mi><mo>=</mo><mn>1</mn><msup><mrow><mn>0</mn></mrow><mrow><mn>50</mn></mrow></msup></mrow></math></span> blue collinear points separated by unit distance. We prove that the statement holds with <span><math><mrow><mi>m</mi><mo>=</mo><mn>1177</mn></mrow></math></span>. We show similar results with different distances between the points.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"125 ","pages":"Article 104105"},"PeriodicalIF":1.0,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136447","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":"Repeatedly applying the Combinatorial Nullstellensatz for Zero-sum Grids to Martin Gardner’s minimum no-3-in-a-line problem","authors":"Seunghwan Oh , John R. Schmitt , Xianzhi Wang","doi":"10.1016/j.ejc.2024.104095","DOIUrl":"10.1016/j.ejc.2024.104095","url":null,"abstract":"<div><div>A 1976 question of Martin Gardner asks for the minimum size of a placement of queens on an <span><math><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></math></span> chessboard that is maximal with respect to the property of ‘no-3-in-a-line’. The work of Cooper, Pikhurko, Schmitt and Warrington showed that this number is at least <span><math><mi>n</mi></math></span> in the cases that <span><math><mrow><mi>n</mi><mo>⁄</mo><mo>≡</mo><mn>3</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>4</mn><mo>)</mo></mrow></mrow></math></span>, and at least <span><math><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span> in the case that <span><math><mrow><mi>n</mi><mo>≡</mo><mn>3</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>4</mn><mo>)</mo></mrow></mrow></math></span>. When <span><math><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow></math></span> is odd, Gardner conjectured the lower bound to be <span><math><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></math></span>. We prove this conjecture in the case that <span><math><mrow><mi>n</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>4</mn><mo>)</mo></mrow></mrow></math></span>. The proof relies heavily on a recent advancement to the Combinatorial Nullstellensatz for zero-sum grids due to Bogdan Nica.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"125 ","pages":"Article 104095"},"PeriodicalIF":1.0,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136074","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":"Classification of countable 2-colored ultrahomogeneous graphs where each color class forms a disjoint union of cliques","authors":"Sofia Brenner, Irene Heinrich","doi":"10.1016/j.ejc.2024.104093","DOIUrl":"10.1016/j.ejc.2024.104093","url":null,"abstract":"<div><div>We classify the countable ultrahomogeneous 2-vertex-colored graphs in which the color classes form disjoint unions of cliques. This generalizes work by Jenkinson et. al. (2012), Lockett and Truss (2014) as well as Rose (2011) on ultrahomogeneous <span><math><mi>n</mi></math></span>-graphs. As the key aspect in such a classification, we identify a concept called piecewise ultrahomogeneity. We prove that there are two specific graphs whose occurrence essentially dictates whether a graph is piecewise ultrahomogeneous, and we exploit this fact to prove the classification.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"125 ","pages":"Article 104093"},"PeriodicalIF":1.0,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136068","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}
Susanna Fishel , Jessica Gatica , Luc Lapointe , María Elena Pinto
{"title":"Fundamental quasisymmetric functions in superspace","authors":"Susanna Fishel , Jessica Gatica , Luc Lapointe , María Elena Pinto","doi":"10.1016/j.ejc.2024.104096","DOIUrl":"10.1016/j.ejc.2024.104096","url":null,"abstract":"<div><div>The fundamental quasisymmetric functions in superspace are a generalization of the fundamental quasisymmetric functions involving anticommuting variables. We obtain the action of the product, coproduct, and antipode on the fundamental quasisymmetric functions in superspace. We also extend to superspace the well known expansion of the Schur functions in terms of fundamental quasisymmetric functions.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"125 ","pages":"Article 104096"},"PeriodicalIF":1.0,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136073","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":"Coloring zonotopal quadrangulations of the projective space","authors":"Masahiro Hachimori , Atsuhiro Nakamoto , Kenta Ozeki","doi":"10.1016/j.ejc.2024.104089","DOIUrl":"10.1016/j.ejc.2024.104089","url":null,"abstract":"<div><div>A quadrangulation on a surface <span><math><msup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is a map of a simple graph on <span><math><msup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> such that each 2-dimensional face is quadrangular. Youngs proved that every quadrangulation on the projective plane <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is either bipartite or 4-chromatic. It is a surprising result since every quadrangulation on an orientable surface with sufficiently high edge-width is 3-colorable. Kaiser and Stehlík defined a <span><math><mi>d</mi></math></span>-dimensional quadrangulation on the <span><math><mi>d</mi></math></span>-dimensional projective space <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> for any <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, and proved that any such quadrangulation has chromatic number at least <span><math><mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow></math></span> if it is not bipartite. In this paper, we define another kind of <span><math><mi>d</mi></math></span>-dimensional quadrangulations of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> for any <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, and prove that such a quadrangulation <span><math><mi>Q</mi></math></span> is always 4-chromatic if <span><math><mi>Q</mi></math></span> is non-bipartite and satisfies a special geometric condition related to a zonotopal tiling of a <span><math><mi>d</mi></math></span>-dimensional zonotope.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"125 ","pages":"Article 104089"},"PeriodicalIF":1.0,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142759709","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":"Rectangulotopes","authors":"Jean Cardinal , Vincent Pilaud","doi":"10.1016/j.ejc.2024.104090","DOIUrl":"10.1016/j.ejc.2024.104090","url":null,"abstract":"<div><div>Rectangulations are decompositions of a square into finitely many axis-aligned rectangles. We describe realizations of <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-dimensional polytopes associated with two combinatorial families of rectangulations composed of <span><math><mi>n</mi></math></span> rectangles. They are defined as quotientopes of natural lattice congruences on the weak Bruhat order on permutations in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and their skeleta are flip graphs on rectangulations. We give simple vertex and facet descriptions of these polytopes, in particular elementary formulas for computing the coordinates of the vertex corresponding to each rectangulation, in the spirit of J.-L. Loday’s realization of the associahedron.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"125 ","pages":"Article 104090"},"PeriodicalIF":1.0,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142756585","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}
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