Discrete Mathematics最新文献

{"title":"On 13-crossing-critical graphs with arbitrarily large degrees","authors":"Petr Hliněný,&nbsp;Michal Korbela","doi":"10.1016/j.disc.2024.114347","DOIUrl":"10.1016/j.disc.2024.114347","url":null,"abstract":"<div><div>A recent result of Bokal et al. (2022) <span><span>[3]</span></span> proved that the exact minimum value of <em>c</em> such that <em>c</em>-crossing-critical graphs do <em>not</em> have bounded maximum degree is <span><math><mi>c</mi><mo>=</mo><mn>13</mn></math></span>. The key to that result is an inductive construction of a family of 13-crossing-critical graphs with many vertices of arbitrarily high degrees. While the inductive part of the construction is rather easy, it all relies on the fact that a certain 17-vertex base graph has the crossing number 13, which was originally verified only by a machine-readable computer proof. We provide a relatively short self-contained computer-free proof of the latter fact. Furthermore, we subsequently generalize the critical construction in order to provide a definitive answer to another long-standing question of this research area; we prove that for every <span><math><mi>c</mi><mo>≥</mo><mn>13</mn></math></span> and integers <span><math><mi>d</mi><mo>,</mo><mi>q</mi></math></span>, there exists a <em>c</em>-crossing-critical graph with more than <em>q</em> vertices of <em>each</em> of the degrees <span><math><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>d</mi></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114347"},"PeriodicalIF":0.7,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143170010","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":"Regular graphs to induce even periodic Grover walks","authors":"Sho Kubota ,&nbsp;Hiroto Sekido ,&nbsp;Kiyoto Yoshino","doi":"10.1016/j.disc.2024.114345","DOIUrl":"10.1016/j.disc.2024.114345","url":null,"abstract":"<div><div>The interest of this paper is a characterization of graphs that induce periodic Grover walks with given periods. In previous studies, Yoshie has shown that the only graphs that induce odd periodic Grover walks are cycle graphs. However, this problem is largely unsolved for even periods. In this study, we show that regular graphs that induce 2<em>l</em>-periodic Grover walks are also cycle graphs in most cases, where <em>l</em> is an odd integer. The proof uses Galois theory.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114345"},"PeriodicalIF":0.7,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743702","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":"Bivariate P- and Q-polynomial structures of the association schemes based on attenuated spaces","authors":"Pierre-Antoine Bernard ,&nbsp;Nicolas Crampé ,&nbsp;Luc Vinet ,&nbsp;Meri Zaimi ,&nbsp;Xiaohong Zhang","doi":"10.1016/j.disc.2024.114332","DOIUrl":"10.1016/j.disc.2024.114332","url":null,"abstract":"<div><div>The bivariate <em>P</em>- and <em>Q</em>-polynomial structures of association schemes based on attenuated spaces are examined using recurrence and difference relations of the bivariate polynomials which form the eigenvalues of the scheme. These bispectral properties are obtained from contiguity relations of univariate dual <em>q</em>-Hahn and affine <em>q</em>-Krawtchouk polynomials. The bispectral algebra associated to the bivariate polynomials is investigated, as well as the subconstituent algebra of the schemes. The properties of the schemes are compared to those of the non-binary Johnson schemes through a limit.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114332"},"PeriodicalIF":0.7,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743701","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":"Palindromicity of the numerator of a statistical generating function","authors":"Rebecca Bourn ,&nbsp;William Q. Erickson","doi":"10.1016/j.disc.2024.114336","DOIUrl":"10.1016/j.disc.2024.114336","url":null,"abstract":"<div><div>We prove a conjecture of Bourn and Willenbring (2020) regarding the palindromicity and unimodality of a certain family of polynomials <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math></span>. These recursively defined polynomials arise as the numerators of generating functions in the context of the discrete one-dimensional earth mover's distance (EMD). The key to our proof is showing that the defining recursion can be viewed as describing sums of symmetric differences of pairs of Young diagrams; in this setting, palindromicity is equivalent to the preservation of the symmetric difference under the transposition of diagrams. We also observe a connection to recent work by Defant et al. (2024) on the Wiener index of minuscule lattices, which we reinterpret combinatorially to obtain explicit formulas for the coefficients of <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math></span> and for the expected value of the discrete EMD.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114336"},"PeriodicalIF":0.7,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722856","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":"Extending simplicial complexes: Topological and combinatorial properties","authors":"Mohammad Farrokhi D.G. ,&nbsp;Alireza Shamsian ,&nbsp;Ali Akbar Yazdan Pour","doi":"10.1016/j.disc.2024.114335","DOIUrl":"10.1016/j.disc.2024.114335","url":null,"abstract":"<div><div>Given an arbitrary hypergraph <span><math><mi>H</mi></math></span>, we may glue to <span><math><mi>H</mi></math></span> a family of hypergraphs to get a new hypergraph <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> having <span><math><mi>H</mi></math></span> as an induced subhypergraph. In this paper, we introduce three gluing techniques for which the topological and combinatorial properties (such as Cohen-Macaulayness, shellability, vertex-decomposability etc.) of the resulting hypergraph <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is under control in terms of the glued components. This enables us to construct broad classes of simplicial complexes containing a given simplicial complex as induced subcomplex satisfying nice topological and combinatorial properties. Our results will be accompanied with some interesting open problems.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114335"},"PeriodicalIF":0.7,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722824","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":"Latin hypercubes realizing integer partitions","authors":"Diane Donovan,&nbsp;Tara Kemp,&nbsp;James Lefevre","doi":"10.1016/j.disc.2024.114333","DOIUrl":"10.1016/j.disc.2024.114333","url":null,"abstract":"<div><div>For an integer partition <span><math><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>N</mi></math></span>, a 2-realization of this partition is a latin square of order <em>N</em> with disjoint subsquares of orders <span><math><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. The existence of 2-realizations is a partially solved problem posed by Fuchs. In this paper, we extend Fuchs' problem to <em>m</em>-ary quasigroups, or, equivalently, latin hypercubes. We construct latin cubes for some partitions with at most two distinct parts and highlight how the new problem is related to the original.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114333"},"PeriodicalIF":0.7,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142701226","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":"Caps and wickets","authors":"Jakob Führer ,&nbsp;Jozsef Solymosi","doi":"10.1016/j.disc.2024.114334","DOIUrl":"10.1016/j.disc.2024.114334","url":null,"abstract":"<div><div>Let <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span> be a 3-uniform linear hypergraph, i.e. any two edges have at most one vertex common. A special hypergraph, <em>wicket</em>, is formed by three rows and two columns of a <span><math><mn>3</mn><mo>×</mo><mn>3</mn></math></span> point matrix. In this note, we give a new lower bound on the Turán number of wickets using estimates on cap sets. We also show that this problem is closely connected to important questions in additive combinatorics.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114334"},"PeriodicalIF":0.7,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722823","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 k-anti-traceability of oriented graphs","authors":"Bin Chen ,&nbsp;Stefanie Gerke ,&nbsp;Gregory Gutin ,&nbsp;Hui Lei ,&nbsp;Heis Parker-Cox ,&nbsp;Yacong Zhou","doi":"10.1016/j.disc.2024.114331","DOIUrl":"10.1016/j.disc.2024.114331","url":null,"abstract":"<div><div>An oriented path <em>P</em> is called anti-directed if every two consecutive arcs of <em>P</em> have opposite orientations. An oriented graph is called <em>k</em>-anti-traceable if every subdigraph induced by <em>k</em> vertices has a hamiltonian anti-directed path. We introduce and study a conjecture, which claims that for every integer <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> there is a least integer <span><math><mi>f</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> such that each <em>k</em>-anti-traceable oriented graph on <span><math><mi>f</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> vertices has a hamiltonian anti-directed path. We determine <span><math><mi>f</mi><mo>(</mo><mn>2</mn><mo>)</mo><mo>,</mo><mi>f</mi><mo>(</mo><mn>3</mn><mo>)</mo><mo>,</mo><mi>f</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span> and show that every <em>k</em>-anti-traceable oriented graph on sufficiently large number <em>n</em> of vertices admits an anti-directed path that contains all but <span><math><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> vertices.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114331"},"PeriodicalIF":0.7,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142701241","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":"New results on a problem of Alon","authors":"Bin Chen","doi":"10.1016/j.disc.2024.114337","DOIUrl":"10.1016/j.disc.2024.114337","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In 2006, Alon proposed a problem of characterizing all four-tuples &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; such that every digraph on &lt;em&gt;n&lt;/em&gt; vertices of minimum out-degree at least &lt;em&gt;s&lt;/em&gt; contains a subdigraph on &lt;em&gt;m&lt;/em&gt; vertices of minimum out-degree at least &lt;em&gt;d&lt;/em&gt;. He in particular asked whether there exists an absolute constant &lt;em&gt;c&lt;/em&gt; such that every digraph on 2&lt;em&gt;n&lt;/em&gt; vertices of minimum out-degree at least &lt;em&gt;s&lt;/em&gt; contains a subdigraph on &lt;em&gt;n&lt;/em&gt; vertices of minimum out-degree at least &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;? Recently, Steiner resolved this case in the negative by showing that for arbitrarily large &lt;em&gt;n&lt;/em&gt;, there exists a tournament on 2&lt;em&gt;n&lt;/em&gt; vertices of minimum out-degree &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, in which the minimum out-degree of every subdigraph on &lt;em&gt;n&lt;/em&gt; vertices is at most &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;In this paper, we study the above problem and present two new results. The first result is that for arbitrary large &lt;em&gt;n&lt;/em&gt; and any integer &lt;span&gt;&lt;math&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, there exists a digraph on &lt;em&gt;αn&lt;/em&gt; vertices of minimum out-degree &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; satisfying that the minimum out-degree of every subdigraph on &lt;em&gt;n&lt;/em&gt; vertices is at most &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. The second result is that for arbitrary large &lt;em&gt;n&lt;/em&gt; and any &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, there exists a digraph on 2&lt;em&gt;n&lt;/em&gt; vertices of girth &lt;em&gt;r&lt;/em&gt; and minimum out-degree &lt;em&gt;s&lt;/em&gt; satisfying that the minimum out-degree of every subdigraph on &lt;em&gt;n&lt;/em&gt; vertices is at most &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; if &lt;em&gt;r&lt;/em&gt; is odd, and is at most &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mo&gt;(&lt;/mo","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114337"},"PeriodicalIF":0.7,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142701227","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":"The complement of the Djoković-Winkler relation","authors":"Marc Hellmuth ,&nbsp;Bruno J. Schmidt ,&nbsp;Guillaume E. Scholz ,&nbsp;Sandhya Thekkumpadan Puthiyaveedu","doi":"10.1016/j.disc.2024.114328","DOIUrl":"10.1016/j.disc.2024.114328","url":null,"abstract":"<div><div>The Djoković-Winkler relation Θ is a binary relation defined on the edge set of a given graph that is based on the distances of certain vertices and which plays a prominent role in graph theory. In this paper, we explore the relatively uncharted “reflexive complement” <span><math><mover><mrow><mi>Θ</mi></mrow><mo>‾</mo></mover></math></span> of Θ, where <span><math><mo>(</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo>)</mo><mo>∈</mo><mover><mrow><mi>Θ</mi></mrow><mo>‾</mo></mover></math></span> if and only if <span><math><mi>e</mi><mo>=</mo><mi>f</mi></math></span> or <span><math><mo>(</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo>)</mo><mo>∉</mo><mi>Θ</mi></math></span> for edges <em>e</em> and <em>f</em>. We establish the relationship between <span><math><mover><mrow><mi>Θ</mi></mrow><mo>‾</mo></mover></math></span> and the set <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>e</mi><mi>f</mi></mrow></msub></math></span>, comprising the distances between the vertices of <em>e</em> and <em>f</em> and shed some light on the intricacies of its transitive closure <span><math><msup><mrow><mover><mrow><mi>Θ</mi></mrow><mo>‾</mo></mover></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. Notably, we demonstrate that <span><math><msup><mrow><mover><mrow><mi>Θ</mi></mrow><mo>‾</mo></mover></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> exhibits multiple equivalence classes only within a restricted subclass of complete multipartite graphs. In addition, we characterize non-trivial relations <em>R</em> that coincide with <span><math><mover><mrow><mi>Θ</mi></mrow><mo>‾</mo></mover></math></span> as those where the graph representation is disconnected, with each connected component being the (join of) Cartesian product of complete graphs. The latter results imply, somewhat surprisingly, that knowledge about the distances between vertices is not required to determine <span><math><msup><mrow><mover><mrow><mi>Θ</mi></mrow><mo>‾</mo></mover></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. Moreover, <span><math><msup><mrow><mover><mrow><mi>Θ</mi></mrow><mo>‾</mo></mover></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> has either exactly one or three equivalence classes.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114328"},"PeriodicalIF":0.7,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142701173","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}