European Journal of Combinatorics最新文献

{"title":"Precoloring extension of Vizing’s Theorem for multigraphs","authors":"Yan Cao ,&nbsp;Guantao Chen ,&nbsp;Guangming Jing ,&nbsp;Xuli Qi ,&nbsp;Songling Shan","doi":"10.1016/j.ejc.2024.104037","DOIUrl":"10.1016/j.ejc.2024.104037","url":null,"abstract":"<div><p>Let <span><math><mi>G</mi></math></span> be a graph with maximum degree <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and maximum multiplicity <span><math><mrow><mi>μ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Vizing and Gupta, independently, proved in the 1960s that the chromatic index of <span><math><mi>G</mi></math></span> is at most <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>μ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. The distance between two edges <span><math><mi>e</mi></math></span> and <span><math><mi>f</mi></math></span> in <span><math><mi>G</mi></math></span> is the length of a shortest path connecting an endvertex of <span><math><mi>e</mi></math></span> and an endvertex of <span><math><mi>f</mi></math></span>. A distance-<span><math><mi>t</mi></math></span> matching is a set of edges having pairwise distance at least <span><math><mi>t</mi></math></span>. Albertson and Moore conjectured that if <span><math><mi>G</mi></math></span> is a simple graph, using the palette <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn><mo>}</mo></mrow></math></span>, any precoloring on a distance-3 matching can be extended to a proper edge coloring of <span><math><mi>G</mi></math></span>. Edwards et al. proposed the following stronger conjecture: For any graph <span><math><mi>G</mi></math></span>, using the palette <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>μ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>}</mo></mrow></math></span>, any precoloring on a distance-2 matching can be extended to a proper edge coloring of <span><math><mi>G</mi></math></span>. Girão and Kang verified the conjecture of Edwards et al. for distance-9 matchings. In this paper, we improve the required distance from 9 to 3 for multigraphs <span><math><mi>G</mi></math></span> with <span><math><mrow><mi>μ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mn>2</mn></mrow></math></span>.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"122 ","pages":"Article 104037"},"PeriodicalIF":1.0,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141950562","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":"Graphs with no even holes and no sector wheels are the union of two chordal graphs","authors":"Tara Abrishami ,&nbsp;Eli Berger ,&nbsp;Maria Chudnovsky ,&nbsp;Shira Zerbib","doi":"10.1016/j.ejc.2024.104035","DOIUrl":"10.1016/j.ejc.2024.104035","url":null,"abstract":"<div><p>Sivaraman (2020) conjectured that if <span><math><mi>G</mi></math></span> is a graph with no induced even cycle then there exist sets <span><math><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> satisfying <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> such that the induced graphs <span><math><mrow><mi>G</mi><mrow><mo>[</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>]</mo></mrow></mrow></math></span> and <span><math><mrow><mi>G</mi><mrow><mo>[</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>]</mo></mrow></mrow></math></span> are both chordal. We prove this conjecture in the special case where <span><math><mi>G</mi></math></span> contains no sector wheel, namely, a pair <span><math><mrow><mo>(</mo><mi>H</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></math></span> where <span><math><mi>H</mi></math></span> is an induced cycle of <span><math><mi>G</mi></math></span> and <span><math><mi>w</mi></math></span> is a vertex in <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>∖</mo><mi>V</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> such that <span><math><mrow><mi>N</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>∩</mo><mi>H</mi></mrow></math></span> is either <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> or a path with at least three vertices.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"122 ","pages":"Article 104035"},"PeriodicalIF":1.0,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141950563","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":"Combinatorial generation via permutation languages. VI. Binary trees","authors":"Petr Gregor ,&nbsp;Torsten Mütze ,&nbsp;Namrata","doi":"10.1016/j.ejc.2024.104020","DOIUrl":"10.1016/j.ejc.2024.104020","url":null,"abstract":"<div><p>In this paper we propose a notion of pattern avoidance in binary trees that generalizes the avoidance of contiguous tree patterns studied by Rowland and non-contiguous tree patterns studied by Dairyko, Pudwell, Tyner, and Wynn. Specifically, we propose algorithms for generating different classes of binary trees that are characterized by avoiding one or more of these generalized patterns. This is achieved by applying the recent Hartung–Hoang–Mütze–Williams generation framework, by encoding binary trees via permutations. In particular, we establish a one-to-one correspondence between tree patterns and certain mesh permutation patterns. We also conduct a systematic investigation of all tree patterns on at most 5 vertices, and we establish bijections between pattern-avoiding binary trees and other combinatorial objects, in particular pattern-avoiding lattice paths and set partitions.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"122 ","pages":"Article 104020"},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001057/pdfft?md5=0f21619605d14f0f464a0bba55039445&pid=1-s2.0-S0195669824001057-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141729129","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":"Subdivisions in dicritical digraphs with large order or digirth","authors":"Lucas Picasarri-Arrieta,&nbsp;Clément Rambaud","doi":"10.1016/j.ejc.2024.104022","DOIUrl":"10.1016/j.ejc.2024.104022","url":null,"abstract":"<div><p>Aboulker et al. proved that a digraph with large enough dichromatic number contains any fixed digraph as a subdivision. The dichromatic number of a digraph is the smallest order of a partition of its vertex set into acyclic induced subdigraphs. A digraph is dicritical if the removal of any arc or vertex decreases its dichromatic number. In this paper we give sufficient conditions on a dicritical digraph of large order or large directed girth to contain a given digraph as a subdivision. In particular, we prove that (i) for every integers <span><math><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></math></span>, large enough dicritical digraphs with dichromatic number <span><math><mi>k</mi></math></span> contain an orientation of a cycle with at least <span><math><mi>ℓ</mi></math></span> vertices; (ii) there are functions <span><math><mrow><mi>f</mi><mo>,</mo><mi>g</mi></mrow></math></span> such that for every subdivision <span><math><msup><mrow><mi>F</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span> of a digraph <span><math><mi>F</mi></math></span>, digraphs with directed girth at least <span><math><mrow><mi>f</mi><mrow><mo>(</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>∗</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> and dichromatic number at least <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> contain a subdivision of <span><math><msup><mrow><mi>F</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>, and if <span><math><mi>F</mi></math></span> is a tree, then <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span>; (iii) there is a function <span><math><mi>f</mi></math></span> such that for every subdivision <span><math><msup><mrow><mi>F</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span> of <span><math><mrow><mi>T</mi><msub><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span> (the transitive tournament on three vertices), digraphs with directed girth at least <span><math><mrow><mi>f</mi><mrow><mo>(</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>∗</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> and minimum out-degree at least 2 contain <span><math><msup><mrow><mi>F</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span> as a subdivision.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"122 ","pages":"Article 104022"},"PeriodicalIF":1.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141630308","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":"Enumerating regions of Shi arrangements per Weyl cone","authors":"Aram Dermenjian ,&nbsp;Eleni Tzanaki","doi":"10.1016/j.ejc.2024.104002","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.104002","url":null,"abstract":"<div><p>Given a Shi arrangement <span><math><msub><mrow><mo>Shi</mo></mrow><mrow><mi>Φ</mi></mrow></msub></math></span>, it is well-known that the total number of regions is counted by the parking number of type <span><math><mi>Φ</mi></math></span> and the total number of regions in the dominant cone is given by the Catalan number of type <span><math><mi>Φ</mi></math></span>. In the case of the latter, in Shi (1997), Shi gave a bijection between antichains in the root poset of <span><math><mi>Φ</mi></math></span> and the regions in the dominant cone. This result was later extended by Armstrong, Reiner and Rhoades in Armstrong et al. (2015) where they gave a bijection between the number of regions contained in an arbitrary Weyl cone <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>w</mi></mrow></msub></math></span> in <span><math><msub><mrow><mo>Shi</mo></mrow><mrow><mi>Φ</mi></mrow></msub></math></span> and certain subposets of the root poset. In this article we expand on these results by giving a determinantal formula for the precise number of regions in <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>w</mi></mrow></msub></math></span> using paths in certain digraphs related to Shi diagrams.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"122 ","pages":"Article 104002"},"PeriodicalIF":1.0,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000878/pdfft?md5=ab820f1a5561b5235bb9cad9f35e2215&pid=1-s2.0-S0195669824000878-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141539893","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":"Skew symplectic and orthogonal characters through lattice paths","authors":"Seamus P. Albion ,&nbsp;Ilse Fischer ,&nbsp;Hans Höngesberg ,&nbsp;Florian Schreier-Aigner","doi":"10.1016/j.ejc.2024.104000","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.104000","url":null,"abstract":"<div><p>The skew Schur functions admit many determinantal expressions. Chief among them are the (dual) Jacobi–Trudi formula and the Lascoux–Pragacz formula, the latter being a skew analogue of the Giambelli identity. Comparatively, the skew characters of the symplectic and orthogonal groups, also known as the skew symplectic and orthogonal Schur functions, have received less attention in this direction. We establish analogues of the dual Jacobi–Trudi and Lascoux–Pragacz formulae for these characters. Our approach is entirely combinatorial, being based on lattice path descriptions of the tableaux models of Koike and Terada. Ordinary Jacobi–Trudi formulae are then derived in an algebraic manner from their duals.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"122 ","pages":"Article 104000"},"PeriodicalIF":1.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000854/pdfft?md5=27fd60792f502dd428590e06520747af&pid=1-s2.0-S0195669824000854-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141539894","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":"Linear extensions and continued fractions","authors":"Swee Hong Chan ,&nbsp;Igor Pak","doi":"10.1016/j.ejc.2024.104018","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.104018","url":null,"abstract":"<div><p>We introduce several new constructions of finite posets with the number of linear extensions given by generalized continued fractions. We apply our results to the problem of the minimum number of elements needed for a poset with a given number of linear extensions.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"122 ","pages":"Article 104018"},"PeriodicalIF":1.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001033/pdfft?md5=54c557aaf8c82af5cc506fbf46b6ca94&pid=1-s2.0-S0195669824001033-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141479934","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":"Ramsey goodness of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" id=\"d1e110\" altimg=\"si29.svg\"><mml:mi>k</mml:mi></mml:math>-uniform paths, or the lack thereof","authors":"Simona Boyadzhiyska, Allan Lo","doi":"10.1016/j.ejc.2024.104021","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.104021","url":null,"abstract":"","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"15 20","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141846397","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":"Links in orthoplicial Apollonian packings","authors":"Jorge L. Ramírez Alfonsín ,&nbsp;Iván Rasskin","doi":"10.1016/j.ejc.2024.104017","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.104017","url":null,"abstract":"<div><p>In this paper, we establish a connection between Apollonian packings and knot theory. We introduce new representations of links realized in the tangency graph of the regular crystallographic sphere packings. Particularly, we prove that any algebraic link can be realized in the cubic section of the orthoplicial Apollonian packing. We use these representations to improve the upper bound on the ball number of an infinite family of alternating algebraic links. Furthermore, the later allow us to reinterpret the correspondence of rational tangles and rational numbers and to reveal geometrically primitive solutions for the Diophantine equation <span><math><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>+</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>+</mo><msup><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>=</mo><mn>2</mn><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"122 ","pages":"Article 104017"},"PeriodicalIF":1.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141479935","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":"Asymptotics of local face distributions and the face distribution of the complete graph","authors":"Jesse Campion Loth","doi":"10.1016/j.ejc.2024.104019","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.104019","url":null,"abstract":"<div><p>We are interested in the distribution of the number of faces across all the <span><math><mrow><mn>2</mn><mo>−</mo></mrow></math></span>cell embeddings of a graph, which is equivalent to the distribution of genus by Euler’s formula. In order to study this distribution, we consider the local distribution of faces at a single vertex. We show an asymptotic uniformity on this local face distribution which holds for any graph with large vertex degrees.</p><p>We use this to study the usual face distribution of the complete graph. We show that in this case, the local face distribution determines the face distribution for almost all of the whole graph. We use this result to show that a portion of the complete graph of size <span><math><mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mo>|</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>|</mo></mrow></mrow></math></span> has the same face distribution as the set of all permutations, up to parity. Along the way, we prove new character bounds and an asymptotic uniformity on conjugacy class products.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"122 ","pages":"Article 104019"},"PeriodicalIF":1.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001045/pdfft?md5=fc1ded09367fd9c944d4c7c3dd000f19&pid=1-s2.0-S0195669824001045-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141479937","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}