Discrete Mathematics最新文献

{"title":"N-ary groups of panmagic permutations from the Post coset theorem","authors":"Sergiy Koshkin ,&nbsp;Jaeho Lee","doi":"10.1016/j.disc.2025.114467","DOIUrl":"10.1016/j.disc.2025.114467","url":null,"abstract":"<div><div>Panmagic permutations are permutations whose matrices are panmagic squares, better known as maximal configurations of non-attacking queens on a toroidal chessboard. Some of them, affine panmagic permutations, can be conveniently described by linear formulas of modular arithmetic, and we show that their sets are a generalization of groups with <em>N</em>-ary multiplication instead of binary one. With the help of the Post coset theorem, we identify panmagic <em>N</em>-ary groups as cosets of the dihedral subgroup and its extensions in the group of all affine permutations. We also investigate decomposition of panmagic permutations into disjoint cycles and find many connections with classical topics of number theory and combinatorics: square-free numbers, <span><math><mn>4</mn><mi>k</mi><mo>+</mo><mn>1</mn></math></span> primes, quadratic residues, cycle indices from Polya counting, and linear congruential generators.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114467"},"PeriodicalIF":0.7,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143563799","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 Galois LCD codes and LCPs of codes over mixed alphabets","authors":"Leijo Jose,&nbsp;Anuradha Sharma","doi":"10.1016/j.disc.2025.114465","DOIUrl":"10.1016/j.disc.2025.114465","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;R&lt;/span&gt; be a finite commutative chain ring with the maximal ideal &lt;span&gt;&lt;math&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of nilpotency index &lt;span&gt;&lt;math&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, and let &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; for some positive integer &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, we study and characterize Galois &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt;-LCD codes of an arbitrary block-length. We show that each weakly-free &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt;-linear code is monomially equivalent to a Galois &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt;-LCD code when &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, while it is monomially equivalent to a Euclidean &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt;-LCD code when &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. We also obtain enumeration formulae for all Euclidean and Hermitian &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt;-LCD codes of an arbitrary block-length. With the help of these enumeration formulae, we classify all Euclidean &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-LCD codes and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-LCD codes of block-lengths &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and all Hermitian &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-LCD codes of block-lengths &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114465"},"PeriodicalIF":0.7,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143563865","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":"Every nonsymmetric 4-class association scheme can be generated by a digraph","authors":"Yuefeng Yang","doi":"10.1016/j.disc.2025.114478","DOIUrl":"10.1016/j.disc.2025.114478","url":null,"abstract":"<div><div>A (di)graph Γ generates a commutative association scheme <span><math><mi>X</mi></math></span> if and only if the adjacency matrix of Γ generates the Bose-Mesner algebra of <span><math><mi>X</mi></math></span>. In <span><span>[18, Theorem 1.1]</span></span>, Monzillo and Penjić proved that, except for amorphic symmetric association schemes, every 3-class association scheme can be generated by the adjacency matrix of a (di)graph. In this paper, we characterize when a commutative association scheme with exactly one pair of nonsymmetric relations can be generated by a digraph under certain assumptions. As an application, we show that each nonsymmetric 4-class association scheme can be generated by a digraph.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114478"},"PeriodicalIF":0.7,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143563800","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 Steinerberger curvature and graph distance matrices","authors":"Wei-Chia Chen ,&nbsp;Mao-Pei Tsui","doi":"10.1016/j.disc.2025.114475","DOIUrl":"10.1016/j.disc.2025.114475","url":null,"abstract":"<div><div>Steinerberger proposed a notion of curvature on graphs involving the graph distance matrix (J. Graph Theory, 2023). We show that nonnegative curvature is almost preserved under three graph operations. We characterize the distance matrix and its null space after adding an edge between two graphs. Let <em>D</em> be the graph distance matrix and <strong>1</strong> be the all-one vector. We provide a way to construct graphs so that the linear system <span><math><mi>D</mi><mi>x</mi><mo>=</mo><mn>1</mn></math></span> does not have a solution.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114475"},"PeriodicalIF":0.7,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143563864","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":"Proof of a conjecture on connectivity keeping odd paths in k-connected bipartite graphs","authors":"Qing Yang,&nbsp;Yingzhi Tian","doi":"10.1016/j.disc.2025.114476","DOIUrl":"10.1016/j.disc.2025.114476","url":null,"abstract":"<div><div>Luo, Tian and Wu (2022) conjectured that for any tree <em>T</em> with bipartition <em>X</em> and <em>Y</em>, every <em>k</em>-connected bipartite graph <em>G</em> with minimum degree at least <span><math><mi>k</mi><mo>+</mo><mi>t</mi></math></span>, where <span><math><mi>t</mi><mo>=</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><mo>|</mo><mi>X</mi><mo>|</mo><mo>,</mo><mo>|</mo><mi>Y</mi><mo>|</mo><mo>}</mo></math></span>, contains a tree <span><math><msup><mrow><mi>T</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≅</mo><mi>T</mi></math></span> such that <span><math><mi>G</mi><mo>−</mo><mi>V</mi><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></math></span> is still <em>k</em>-connected. Note that <span><math><mi>t</mi><mo>=</mo><mo>⌈</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></math></span> when the tree <em>T</em> is the path with order <em>m</em>. In this paper, we prove that every <em>k</em>-connected bipartite graph <em>G</em> with minimum degree at least <span><math><mi>k</mi><mo>+</mo><mo>⌈</mo><mfrac><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></math></span> contains a path <em>P</em> of order <em>m</em> such that <span><math><mi>G</mi><mo>−</mo><mi>V</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span> remains <em>k</em>-connected. This shows that the conjecture is true for paths with odd order. For paths with even order, the minimum degree bound in this paper is the bound in the conjecture plus one.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114476"},"PeriodicalIF":0.7,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143563866","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":"Induced saturation for complete bipartite posets","authors":"Dingyuan Liu","doi":"10.1016/j.disc.2025.114462","DOIUrl":"10.1016/j.disc.2025.114462","url":null,"abstract":"<div><div>Given <span><math><mi>s</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>N</mi></math></span>, a complete bipartite poset <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span> is a poset whose Hasse diagram consists of <em>s</em> pairwise incomparable vertices in the upper layer and <em>t</em> pairwise incomparable vertices in the lower layer, such that every vertex in the upper layer is larger than all vertices in the lower layer. A family <span><math><mi>F</mi><mo>⊆</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></msup></math></span> is called induced <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span>-saturated if <span><math><mo>(</mo><mi>F</mi><mo>,</mo><mo>⊆</mo><mo>)</mo></math></span> contains no induced copy of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span>, whereas adding any set from <span><math><msup><mrow><mn>2</mn></mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></msup><mo>﹨</mo><mi>F</mi></math></span> to <span><math><mi>F</mi></math></span> creates an induced <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span>. Let <span><math><msup><mrow><mi>sat</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>)</mo></math></span> denote the smallest size of an induced <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span>-saturated family <span><math><mi>F</mi><mo>⊆</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></msup></math></span>. It was conjectured that <span><math><msup><mrow><mi>sat</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>)</mo></math></span> is superlinear in <em>n</em> for certain values of <em>s</em> and <em>t</em>. In this paper, we show that <span><math><msup><mrow><mi>sat</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for all fixed <span><math><mi>s</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>N</mi></math></span>. Moreover, we prove a linear lower bound on <span><math><msup><mrow><mi>sat</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> for a large class of posets <span><math><mi>P</mi></math></span>, particularly for <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mn>2</mn></mrow></msub></math></span> with <span><math><mi>s</mi><mo>∈</mo><mi>N</mi></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114462"},"PeriodicalIF":0.7,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143549779","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":"The Ramsey numbers for certain large trees of order n with maximum degree at most n − 6 versus the wheel of order nine","authors":"Thomas Britz ,&nbsp;Zhi Yee Chng ,&nbsp;Kok Bin Wong","doi":"10.1016/j.disc.2025.114461","DOIUrl":"10.1016/j.disc.2025.114461","url":null,"abstract":"<div><div>For a fixed positive integer <span><math><mi>k</mi><mo>≥</mo><mn>5</mn></math></span>, the Ramsey numbers <span><math><mi>R</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>W</mi></mrow><mrow><mn>8</mn></mrow></msub><mo>)</mo></math></span> are determined for the tree <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of sufficiently large order <em>n</em> and maximum degree <span><math><mi>Δ</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mn>1</mn></math></span>. This result provides a partial proof for the conjecture, due to Chen, Zhang and Zhang and to Hafidh and Baskoro, that <span><math><mi>R</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></math></span> for each tree <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of order <span><math><mi>n</mi><mo>≥</mo><mi>m</mi><mo>−</mo><mn>1</mn></math></span> with <span><math><mi>Δ</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>≤</mo><mi>n</mi><mo>−</mo><mi>m</mi><mo>+</mo><mn>2</mn></math></span> when <span><math><mi>m</mi><mo>≥</mo><mn>4</mn></math></span> is even, for the case when <span><math><mi>m</mi><mo>=</mo><mn>8</mn></math></span> and <em>n</em> is sufficiently large.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114461"},"PeriodicalIF":0.7,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143535040","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 large minimal blocker for 123-avoiding permutations","authors":"Yaroslav Shitov","doi":"10.1016/j.disc.2025.114463","DOIUrl":"10.1016/j.disc.2025.114463","url":null,"abstract":"&lt;div&gt;&lt;div&gt;A set &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&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;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;mo&gt;×&lt;/mo&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;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is a &lt;em&gt;blocker of&lt;/em&gt; a subset &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; of the symmetric group &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; if every permutation &lt;span&gt;&lt;math&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; allows an index &lt;em&gt;i&lt;/em&gt; with &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Bennett, Brualdi and Cao conjectured that &lt;span&gt;&lt;math&gt;&lt;mo&gt;⌈&lt;/mo&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;mo&gt;)&lt;/mo&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;⌉&lt;/mo&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mo&gt;⌊&lt;/mo&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;mo&gt;)&lt;/mo&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;⌋&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is an upper bound for the sizes of the inclusion minimal blockers of the family of 123-&lt;em&gt;avoiding&lt;/em&gt; permutations, which are those &lt;span&gt;&lt;math&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; for which &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; has no increasing subsequence of the length three. We show that&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/m","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114463"},"PeriodicalIF":0.7,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143535042","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":"Several classes of minimal linear codes from vectorial Boolean functions and p-ary functions","authors":"Wengang Jin,&nbsp;Kangquan Li,&nbsp;Longjiang Qu","doi":"10.1016/j.disc.2025.114464","DOIUrl":"10.1016/j.disc.2025.114464","url":null,"abstract":"<div><div>Minimal linear codes are widely used in secret sharing schemes and secure two-party computation. Most of the minimal linear codes constructed satisfy the Ashikhmin-Barg (AB for short) condition. However, up to now, only a small classes of minimal linear codes violating the AB condition have been presented in the literature. In this paper, we are devoted to constructing more classes of minimal linear codes over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> that violate the AB condition and have new parameters. First, we provide several classes of minimal linear codes violating the AB condition from vectorial Boolean functions and determine their weight distributions. Then, we obtain new <em>p</em>-ary functions over the finite fields <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> with <em>p</em> an odd prime and determine their Walsh spectrum distributions. Finally, the resulted <em>p</em>-ary functions are employed to construct several classes of linear codes with two to four weights. In these codes, one class is minimal and violates the AB condition, and two classes satisfy the AB condition.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114464"},"PeriodicalIF":0.7,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143535041","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":"Induced matching vs edge open packing: Trees and product graphs","authors":"Boštjan Brešar ,&nbsp;Tanja Dravec ,&nbsp;Jaka Hedžet ,&nbsp;Babak Samadi","doi":"10.1016/j.disc.2025.114458","DOIUrl":"10.1016/j.disc.2025.114458","url":null,"abstract":"<div><div>Given a graph <em>G</em>, the maximum size of an induced subgraph of <em>G</em> each component of which is a star is called the edge open packing number, <span><math><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mi>o</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, of <em>G</em>. Similarly, the maximum size of an induced subgraph of <em>G</em> each component of which is the star <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub></math></span> is the induced matching number, <span><math><msub><mrow><mi>ν</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, of <em>G</em>. While the inequality <span><math><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mi>o</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> clearly holds for all graphs <em>G</em>, we provide a structural characterization of those trees that attain the equality. We prove that the induced matching number of the lexicographic product <span><math><mi>G</mi><mo>∘</mo><mi>H</mi></math></span> of arbitrary two graphs <em>G</em> and <em>H</em> equals <span><math><mi>α</mi><mo>(</mo><mi>G</mi><mo>)</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo></math></span>. By similar techniques, we prove sharp lower and upper bounds on the edge open packing number of the lexicographic product of graphs, which in particular lead to NP-hardness results in triangular graphs for both invariants studied in this paper. For the direct product <span><math><mi>G</mi><mo>×</mo><mi>H</mi></math></span> of two graphs we provide lower bounds on <span><math><msub><mrow><mi>ν</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>×</mo><mi>H</mi><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mi>o</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>×</mo><mi>H</mi><mo>)</mo></math></span>, both of which are widely sharp. We also present sharp lower bounds for both invariants in the Cartesian and the strong product of two graphs. Finally, we consider the edge open packing number in hypercubes establishing the exact values of <span><math><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mi>o</mi></mrow></msubsup><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> when <em>n</em> is a power of 2, and present a closed formula for the induced matching number of the rooted product of arbitrary two graphs over an arbitrary root vertex.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114458"},"PeriodicalIF":0.7,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143535043","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}