European Journal of Combinatorics最新文献

{"title":"Repeatedly applying the Combinatorial Nullstellensatz for Zero-sum Grids to Martin Gardner’s minimum no-3-in-a-line problem","authors":"Seunghwan Oh ,&nbsp;John R. Schmitt ,&nbsp;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>&gt;</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":"Stack and queue numbers of graphs revisited","authors":"Petr Hliněný,&nbsp;Adam Straka","doi":"10.1016/j.ejc.2024.104094","DOIUrl":"10.1016/j.ejc.2024.104094","url":null,"abstract":"<div><div>A long-standing question of the mutual relation between the stack and queue numbers of a graph, explicitly emphasized by Dujmović and Wood in 2005, was partially answered by Dujmović, Eppstein, Hickingbotham, Morin and Wood in 2022; they proved the existence of a graph family with the queue number at most 4 but unbounded stack number. We give an alternative very short, and still elementary, proof of the same fact.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"129 ","pages":"Article 104094"},"PeriodicalIF":1.0,"publicationDate":"2024-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144588402","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,&nbsp;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}

{"title":"Fundamental quasisymmetric functions in superspace","authors":"Susanna Fishel ,&nbsp;Jessica Gatica ,&nbsp;Luc Lapointe ,&nbsp;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":"A survey of degree-boundedness","authors":"Xiying Du ,&nbsp;Rose McCarty","doi":"10.1016/j.ejc.2024.104092","DOIUrl":"10.1016/j.ejc.2024.104092","url":null,"abstract":"<div><div>Suppose a graph has no large balanced bicliques, but has large minimum degree. Then what can we say about its induced subgraphs? This question motivates the study of degree-boundedness, which is like <span><math><mi>χ</mi></math></span>-boundedness but for minimum degree instead of chromatic number. We survey this area with an eye towards open problems.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"129 ","pages":"Article 104092"},"PeriodicalIF":1.0,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144588351","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 ,&nbsp;Atsuhiro Nakamoto ,&nbsp;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 ,&nbsp;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}

{"title":"On the order of semiregular automorphisms of cubic vertex-transitive graphs","authors":"Marco Barbieri ,&nbsp;Valentina Grazian ,&nbsp;Pablo Spiga","doi":"10.1016/j.ejc.2024.104091","DOIUrl":"10.1016/j.ejc.2024.104091","url":null,"abstract":"<div><div>We prove that, if <span><math><mi>Γ</mi></math></span> is a finite connected cubic vertex-transitive graph, then either there exists a semiregular automorphism of <span><math><mi>Γ</mi></math></span> of order at least 6, or the number of vertices of <span><math><mi>Γ</mi></math></span> is bounded above by an absolute constant.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104091"},"PeriodicalIF":1.0,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142705764","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":"More on rainbow cliques in edge-colored graphs","authors":"Xiao-Chuan Liu ,&nbsp;Danni Peng ,&nbsp;Xu Yang","doi":"10.1016/j.ejc.2024.104088","DOIUrl":"10.1016/j.ejc.2024.104088","url":null,"abstract":"<div><div>In an edge-colored graph <span><math><mi>G</mi></math></span>, a rainbow clique <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is a complete subgraph on <span><math><mi>k</mi></math></span> vertices in which all the edges have distinct colors. Let <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the number of edges and colors in <span><math><mi>G</mi></math></span>, respectively. In this paper, we show that for any <span><math><mrow><mi>ɛ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>, if <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>3</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mn>2</mn><mi>ɛ</mi><mo>)</mo></mrow><mfenced><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mfenced></mrow></math></span> and <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, then for sufficiently large <span><math><mi>n</mi></math></span>, the number of rainbow cliques <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> in <span><math><mi>G</mi></math></span> is <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>.</div><div>We also characterize the extremal graphs <span><math><mi>G</mi></math></span> without a rainbow clique <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, for <span><math><mrow><mi>k</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>5</mn></mrow></math></span>, when <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is maximum.</div><div>Our results not only address existing questions but also complete the findings of Ehard and Mohr (2020).</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104088"},"PeriodicalIF":1.0,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142705765","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 (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}