Serafino Cicerone , Gabriele Di Stefano , Sandi Klavžar , Ismael G. Yero
{"title":"Mutual-visibility problems on graphs of diameter two","authors":"Serafino Cicerone , Gabriele Di Stefano , Sandi Klavžar , Ismael G. Yero","doi":"10.1016/j.ejc.2024.103995","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103995","url":null,"abstract":"<div><p>The mutual-visibility problem in a graph <span><math><mi>G</mi></math></span> asks for the cardinality of a largest set of vertices <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> so that for any two vertices <span><math><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi></mrow></math></span> there is a shortest <span><math><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></math></span>-path <span><math><mi>P</mi></math></span> so that all internal vertices of <span><math><mi>P</mi></math></span> are not in <span><math><mi>S</mi></math></span>. This is also said as <span><math><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></math></span> are visible with respect to <span><math><mi>S</mi></math></span>, or <span><math><mi>S</mi></math></span>-visible for short. Variations of this problem are known, based on the extension of the visibility property of vertices that are in and/or outside <span><math><mi>S</mi></math></span>. Such variations are called total, outer and dual mutual-visibility problems. This work is focused on studying the corresponding four visibility parameters in graphs of diameter two, throughout showing bounds and/or closed formulae for these parameters.</p><p>The mutual-visibility problem in the Cartesian product of two complete graphs is equivalent to (an instance of) the celebrated Zarankiewicz’s problem. Here we study the dual and outer mutual-visibility problem for the Cartesian product of two complete graphs and all the mutual-visibility problems for the direct product of such graphs as well. We also study all the mutual-visibility problems for the line graphs of complete and complete bipartite graphs. As a consequence of this study, we present several relationships between the mentioned problems and some instances of the classical Turán problem. Moreover, we study the visibility problems for cographs and several non-trivial diameter-two graphs of minimum size.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000805/pdfft?md5=8b2737f5ff200dbacddcfc4e58622e8e&pid=1-s2.0-S0195669824000805-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141090829","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}
Toby Aldape , Jingyi Liu , Gregory Pylypovych , Adam Sheffer , Minh-Quan Vo
{"title":"Distinct distances in R3 between quadratic and orthogonal curves","authors":"Toby Aldape , Jingyi Liu , Gregory Pylypovych , Adam Sheffer , Minh-Quan Vo","doi":"10.1016/j.ejc.2024.103993","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103993","url":null,"abstract":"<div><p>We study the minimum number of distinct distances between point sets on two curves in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. Assume that one curve contains <span><math><mi>m</mi></math></span> points and the other <span><math><mi>n</mi></math></span> points. Our main results:</p><p>(a) When the curves are conic sections, we characterize all cases where the number of distances is <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>. This includes new constructions for points on two parabolas, two ellipses, and one ellipse and one hyperbola. In all other cases, the number of distances is <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><mo>min</mo><mrow><mo>{</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>,</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>.</p><p>(b) When the curves are not necessarily algebraic but smooth and contained in perpendicular planes, we characterize all cases where the number of distances is <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>. This includes a surprising new construction of non-algebraic curves that involve logarithms. In all other cases, the number of distances is <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><mo>min</mo><mrow><mo>{</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>,</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141068815","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":"Variations on the Bollobás set-pair theorem","authors":"Gábor Hegedüs , Péter Frankl","doi":"10.1016/j.ejc.2024.103983","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103983","url":null,"abstract":"<div><p>Let <span><math><mi>X</mi></math></span> be an <span><math><mi>n</mi></math></span>-element set. A set-pair system <span><math><mrow><mi>P</mi><mo>=</mo><msub><mrow><mrow><mo>{</mo><mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>}</mo></mrow></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>m</mi></mrow></msub></mrow></math></span> is a collection of pairs of disjoint subsets of <span><math><mi>X</mi></math></span>. It is called skew Bollobás system if <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≠</mo><mo>0̸</mo></mrow></math></span> for all <span><math><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>≤</mo><mi>m</mi></mrow></math></span>. The best possible inequality <span><math><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><mfrac><mrow><mn>1</mn></mrow><mrow><mfenced><mfrac><mrow><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow><mo>+</mo><mrow><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mrow><mrow><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mrow></mfrac></mfenced></mrow></mfrac><mo>≤</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>.</mo></mrow></math></span> is established along with some more results of similar flavor.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000684/pdfft?md5=21ca6f2c126894c4d8c03cff15a181b3&pid=1-s2.0-S0195669824000684-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140902015","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":"An identity of Ramanujan and its combinatorics","authors":"Bernard L.S. Lin , Xiaowei Lin , Lei Zhang","doi":"10.1016/j.ejc.2024.103985","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103985","url":null,"abstract":"<div><p>In this paper, we explore the combinatorics behind an identity recorded in Ramanujan’s lost notebook. We present an interesting result, which not only generalizes two theorems of Bressoud, but also implies a bivariate form of Ramanujan’s original identity.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140902018","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":"Bounds for the number of multidimensional partitions","authors":"Kristina Oganesyan","doi":"10.1016/j.ejc.2024.103982","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103982","url":null,"abstract":"<div><p>We obtain estimates for the number <span><math><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-dimensional integer partitions of a number <span><math><mi>n</mi></math></span>. It is known that the two-sided inequality <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>d</mi></mrow></msup><mo><</mo><mo>log</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo><</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>d</mi></mrow></msup></mrow></math></span> is always true and that <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow><mo>></mo><mn>1</mn></mrow></math></span> whenever <span><math><mrow><mo>log</mo><mi>n</mi><mo>></mo><mn>3</mn><mi>d</mi></mrow></math></span>. However, establishing the <span><math><mi>“</mi></math></span>right<span><math><mi>”</mi></math></span> dependence of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> on <span><math><mi>d</mi></math></span> remained an open problem. We show that if <span><math><mi>d</mi></math></span> is sufficiently small with respect to <span><math><mi>n</mi></math></span>, then <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> does not depend on <span><math><mi>d</mi></math></span>, which means that <span><math><mrow><mo>log</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> is up to an absolute constant equal to <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>d</mi></mrow></msup></math></span>. Besides, we provide estimates of <span><math><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> for different ranges of <span><math><mi>d</mi></math></span> in terms of <span><math><mi>n</mi></math></span>, which give the asymptotics of <span><math><mrow><mo>log</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> in each case.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140902016","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":"Two Ramsey problems in blowups of graphs","authors":"António Girão , Robert Hancock","doi":"10.1016/j.ejc.2024.103984","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103984","url":null,"abstract":"<div><p>Given graphs <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span>, we say <span><math><mrow><mi>G</mi><mover><mrow><mo>→</mo></mrow><mrow><mrow><mi>r</mi></mrow></mrow></mover><mi>H</mi></mrow></math></span> if every <span><math><mi>r</mi></math></span>-colouring of the edges of <span><math><mi>G</mi></math></span> contains a monochromatic copy of <span><math><mi>H</mi></math></span>. Let <span><math><mrow><mi>H</mi><mrow><mo>[</mo><mi>t</mi><mo>]</mo></mrow></mrow></math></span> denote the <span><math><mi>t</mi></math></span>-blowup of <span><math><mi>H</mi></math></span>. The blowup Ramsey number <span><math><mrow><mi>B</mi><mrow><mo>(</mo><mi>G</mi><mover><mrow><mo>→</mo></mrow><mrow><mrow><mi>r</mi></mrow></mrow></mover><mi>H</mi><mo>;</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> is the minimum <span><math><mi>n</mi></math></span> such that <span><math><mrow><mi>G</mi><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mover><mrow><mo>→</mo></mrow><mrow><mrow><mi>r</mi></mrow></mrow></mover><mi>H</mi><mrow><mo>[</mo><mi>t</mi><mo>]</mo></mrow></mrow></math></span>. Fox, Luo and Wigderson refined an upper bound of Souza, showing that, given <span><math><mi>G</mi></math></span>, <span><math><mi>H</mi></math></span> and <span><math><mi>r</mi></math></span> such that <span><math><mrow><mi>G</mi><mover><mrow><mo>→</mo></mrow><mrow><mrow><mi>r</mi></mrow></mrow></mover><mi>H</mi></mrow></math></span>, there exist constants <span><math><mrow><mi>a</mi><mo>=</mo><mi>a</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>b</mi><mo>=</mo><mi>b</mi><mrow><mo>(</mo><mi>H</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span> such that for all <span><math><mrow><mi>t</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, <span><math><mrow><mi>B</mi><mrow><mo>(</mo><mi>G</mi><mover><mrow><mo>→</mo></mrow><mrow><mrow><mi>r</mi></mrow></mrow></mover><mi>H</mi><mo>;</mo><mi>t</mi><mo>)</mo></mrow><mo>≤</mo><mi>a</mi><msup><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></math></span>. They conjectured that there exist some graphs <span><math><mi>H</mi></math></span> for which the constant <span><math><mi>a</mi></math></span> depending on <span><math><mi>G</mi></math></span> is necessary. We prove this conjecture by showing that the statement is true in the case of <span><math><mi>H</mi></math></span> being 3-chromatically connected, which in particular includes triangles. On the other hand, perhaps surprisingly, we show that for forests <span><math><mi>F</mi></math></span>, there exists an upper bound for <span><math><mrow><mi>B</mi><mrow><mo>(</mo><mi>G</mi><mover><mrow><mo>→</mo></mrow><mrow><mrow><mi>r</mi></mrow></mrow></mover><mi>F</mi><mo>;</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> which is independent of <span><math><mi>G</mi></math></span>.</p><p>Second, we show that for any <span><math><mrow><mi>r</mi><mo>,</mo><mi","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000696/pdfft?md5=90c712e911fae05fd1803c79c5bbceb8&pid=1-s2.0-S0195669824000696-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140902017","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":"Monochromatic triangles in the max-norm plane","authors":"Alexander Natalchenko , Arsenii Sagdeev","doi":"10.1016/j.ejc.2024.103977","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103977","url":null,"abstract":"<div><p>For all non-degenerate triangles <span><math><mi>T</mi></math></span>, we determine the minimum number of colors needed to color the plane such that no max-norm isometric copy of <span><math><mi>T</mi></math></span> is monochromatic.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140843589","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":"Cycles in Austrian Solitaire","authors":"Philip P. Mummert","doi":"10.1016/j.ejc.2024.103978","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103978","url":null,"abstract":"<div><p>Austrian Solitaire is a variation of Bulgarian Solitaire. It may be described as a card game, a method of asset inventory management, or a discrete dynamical system on integer partitions. We prove that the limit cycles in Austrian Solitaire do not depend on the initial configuration; in other words, each state space is connected. We show that a full Farey sequence completely characterizes these unique (and balanced) cycles.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140842747","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":"Arc-disjoint out- and in-branchings in compositions of digraphs","authors":"J. Bang-Jensen , Y. Wang","doi":"10.1016/j.ejc.2024.103981","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103981","url":null,"abstract":"<div><p>An out-branching <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> (in-branching <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>u</mi></mrow><mrow><mo>−</mo></mrow></msubsup></math></span>) in a digraph <span><math><mi>D</mi></math></span> is a connected spanning subdigraph of <span><math><mi>D</mi></math></span> in which every vertex except the vertex <span><math><mi>u</mi></math></span>, called the root, has in-degree (out-degree) one. A <strong>good</strong><span><math><mi>(u,v)</mi></math></span>-<strong>pair</strong> in <span><math><mi>D</mi></math></span> is a pair of branchings <span><math><mrow><msubsup><mrow><mi>B</mi></mrow><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>v</mi></mrow><mrow><mo>−</mo></mrow></msubsup></mrow></math></span> which have no arc in common. Thomassen proved that it is NP-complete to decide if a digraph has any good pair. A digraph is <strong>semicomplete</strong> if it has no pair of non-adjacent vertices. A <strong>semicomplete composition</strong> is any digraph <span><math><mi>D</mi></math></span> which is obtained from a semicomplete digraph <span><math><mi>S</mi></math></span> by substituting an arbitrary digraph <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>x</mi></mrow></msub></math></span> for each vertex <span><math><mi>x</mi></math></span> of <span><math><mi>S</mi></math></span>.</p><p>Recently the authors of this paper gave a complete classification of semicomplete digraphs which have a good <span><math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></math></span>-pair, where <span><math><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></math></span> are prescribed vertices. They also gave a polynomial algorithm which for a given semicomplete digraph <span><math><mi>D</mi></math></span> and vertices <span><math><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></math></span> of <span><math><mi>D</mi></math></span>, either produces a good <span><math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></math></span>-pair in <span><math><mi>D</mi></math></span> or a certificate that <span><math><mi>D</mi></math></span> has no such pair. In this paper we show how to use the result for semicomplete digraphs to completely solve the problem of characterizing semicomplete compositions which have a good <span><math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></math></span>-pair for given vertices <span><math><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></math></span>. Our solution implies that the problem of deciding the existence of a good <span><math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></math></span>-pair and finding such a pair when it exists is polynomially solvable for all semicomplete compositions. We also completely solve the problem of deciding the existence of a good <span><math><mrow><mo>(</mo><mi>u","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000660/pdfft?md5=9f52566484a640f1db27537236930da5&pid=1-s2.0-S0195669824000660-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140843408","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":"Counting deranged matchings","authors":"Sam Spiro , Erlang Surya","doi":"10.1016/j.ejc.2024.103980","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.103980","url":null,"abstract":"<div><p>Let <span><math><mrow><mi>pm</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denote the number of perfect matchings of a graph <span><math><mi>G</mi></math></span>, and let <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>×</mo><mn>2</mn><mi>n</mi><mo>/</mo><mi>r</mi></mrow></msub></math></span> denote the complete <span><math><mi>r</mi></math></span>-partite graph where each part has size <span><math><mrow><mn>2</mn><mi>n</mi><mo>/</mo><mi>r</mi></mrow></math></span>. Johnson, Kayll, and Palmer conjectured that for any perfect matching <span><math><mi>M</mi></math></span> of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>×</mo><mn>2</mn><mi>n</mi><mo>/</mo><mi>r</mi></mrow></msub></math></span>, we have for <span><math><mrow><mn>2</mn><mi>n</mi></mrow></math></span> divisible by <span><math><mi>r</mi></math></span>\u0000<span><span><span><math><mrow><mfrac><mrow><mi>pm</mi><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>×</mo><mn>2</mn><mi>n</mi><mo>/</mo><mi>r</mi></mrow></msub><mo>−</mo><mi>M</mi><mo>)</mo></mrow></mrow><mrow><mi>pm</mi><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>×</mo><mn>2</mn><mi>n</mi><mo>/</mo><mi>r</mi></mrow></msub><mo>)</mo></mrow></mrow></mfrac><mo>∼</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>r</mi><mo>/</mo><mrow><mo>(</mo><mn>2</mn><mi>r</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow></msup><mo>.</mo></mrow></math></span></span></span>This conjecture can be viewed as a common generalization of counting the number of derangements on <span><math><mi>n</mi></math></span> letters, and of counting the number of deranged matchings of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span>. We prove this conjecture. In fact, we prove the stronger result that if <span><math><mi>R</mi></math></span> is a uniformly random perfect matching of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>×</mo><mn>2</mn><mi>n</mi><mo>/</mo><mi>r</mi></mrow></msub></math></span>, then the number of edges that <span><math><mi>R</mi></math></span> has in common with <span><math><mi>M</mi></math></span> converges to a Poisson distribution with parameter <span><math><mfrac><mrow><mi>r</mi></mrow><mrow><mn>2</mn><mi>r</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span>.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000659/pdfft?md5=c66422b992cbbf0765bc7eba3abcde93&pid=1-s2.0-S0195669824000659-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140650508","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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