Journal of Combinatorial Designs最新文献

{"title":"Toward a Solution of Archdeacon's Conjecture on Integer Heffter Arrays","authors":"Marco Antonio Pellegrini, Tommaso Traetta","doi":"10.1002/jcd.21983","DOIUrl":"https://doi.org/10.1002/jcd.21983","url":null,"abstract":"<p>In this article, we make significant progress on a conjecture proposed by Dan Archdeacon on the existence of integer Heffter arrays <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>;</mo>\u0000 \u0000 <mi>s</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> whenever the necessary conditions hold, that is, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>3</mn>\u0000 \u0000 <mo>⩽</mo>\u0000 \u0000 <mi>s</mi>\u0000 \u0000 <mo>⩽</mo>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>3</mn>\u0000 \u0000 <mo>⩽</mo>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>⩽</mo>\u0000 \u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mi>s</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>≡</mo>\u0000 \u0000 <mn>0</mn>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 8","pages":"310-323"},"PeriodicalIF":0.5,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21983","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144256534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"How to Burn a Latin Square","authors":"Anthony Bonato, Caleb Jones, Trent G. Marbach, Teddy Mishura","doi":"10.1002/jcd.21988","DOIUrl":"https://doi.org/10.1002/jcd.21988","url":null,"abstract":"<p>We investigate the lazy burning process for Latin squares by studying their associated hypergraphs. In lazy burning, a set of vertices in a hypergraph is initially burned, and that burning spreads to neighboring vertices over time via a specified propagation rule. The lazy burning number is the minimum number of initially burned vertices that eventually burns all vertices. The hypergraphs associated with Latin squares include the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-uniform hypergraph, whose vertices and hyperedges correspond to the entries and lines (i.e., sets of rows, columns, or symbols) of the Latin square, respectively, and the 3-uniform hypergraph, which has vertices corresponding to the lines of the Latin square and hyperedges induced by its entries. Using sequences of vertices that together form a vertex cover, we show that for a Latin square of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, the lazy burning number of its <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-uniform hypergraph is bounded below by <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>n</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>3</mn>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and above by <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>n</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>3</mn>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mo>+</mo>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 8","pages":"300-309"},"PeriodicalIF":0.5,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21988","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144255845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Constructions of Optimal Sparse \u0000 \u0000 \u0000 \u0000 r\u0000 \u0000 \u0000 -Disjunct Matrices via Packings","authors":"Liying Yu, Xin Wang, Lijun Ji","doi":"10.1002/jcd.21986","DOIUrl":"https://doi.org/10.1002/jcd.21986","url":null,"abstract":"<div>\u0000 \u0000 <p>Group testing has been widely used in various aspects, and the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-disjunct matrix plays a crucial role in group testing. The original purpose of the group testing is to identify a set of at most <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> positive items from a batch of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>M</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> total items using as fewer tests as possible. In many practical applications, each test can include only a limited number of items and each item can participate in a limited number of tests. In this paper, we use the tools from combinatorial design theory to construct optimal 2-disjunct matrices with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> rows and limited row weight <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>3</mn>\u0000 \u0000 <mo><</mo>\u0000 \u0000 <mi>ρ</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mrow>\u0000 <mo>⌊</mo>\u0000 \u0000 <mfrac>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 \u0000 <mo>⌋</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and optimal 3-disjunct matrices with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> rows and limited row weight <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>4</mn>\u0000 \u0000 <mo><</mo>\u0000 \u0000 <mi>ρ</mi>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 8","pages":"287-299"},"PeriodicalIF":0.5,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144256485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Doubly Orthogonal Equi-Squares and Sliced Orthogonal Arrays","authors":"John Lorch","doi":"10.1002/jcd.21982","DOIUrl":"https://doi.org/10.1002/jcd.21982","url":null,"abstract":"<div>\u0000 \u0000 <p>We introduce doubly orthogonal equi-squares. Using linear algebra over finite fields, we produce large families of mutually <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mi>s</mi>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-doubly orthogonal equi-<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>s</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> squares, and show these are of maximal size when <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>s</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>r</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. These results specialize to the results of Xu, Haaland, and Qian when <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>s</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and the equi-squares are Sudoku Latin squares of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. Further, we show how a collection of mutually <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mi>s</mi>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-doubly orthogonal equi-<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 7","pages":"275-283"},"PeriodicalIF":0.5,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Terwilliger Algebra of the Group Association Scheme of the Symmetric Group \u0000 \u0000 \u0000 \u0000 Sym\u0000 \u0000 (\u0000 7\u0000 )","authors":"Allen Herman, Roghayeh Maleki, Andriaherimanana Sarobidy Razafimahatratra","doi":"10.1002/jcd.21981","DOIUrl":"https://doi.org/10.1002/jcd.21981","url":null,"abstract":"<p>Terwilliger algebras are finite-dimensional semisimple algebras that were first introduced by Paul Terwilliger in 1992 in studies of association schemes and distance-regular graphs. The Terwilliger algebras of the conjugacy class association schemes of the symmetric groups <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>Sym</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>3</mn>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mn>6</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, have been studied and completely determined. The case for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>Sym</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>7</mn>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is computationally much more difficult and has a potential application to find the size of the largest permutation codes of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>Sym</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>7</mn>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> with a minimal distance of at least 4. In this paper, the dimension, the Wedderburn decomposition, and the block dimension decomposition of the Terwilliger algebra of the conjugacy class scheme of the group <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>Sym</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 7","pages":"261-274"},"PeriodicalIF":0.5,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21981","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Directed Oberwolfach Problem With Variable Cycle Lengths: A Recursive Construction","authors":"Suzan Kadri, Mateja Šajna","doi":"10.1002/jcd.21967","DOIUrl":"https://doi.org/10.1002/jcd.21967","url":null,"abstract":"<p>The directed Oberwolfach problem <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mstyle>\u0000 <mspace></mspace>\u0000 \u0000 <mtext>OP</mtext>\u0000 <mspace></mspace>\u0000 </mstyle>\u0000 \u0000 <mo>*</mo>\u0000 </msup>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>m</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>…</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>m</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> asks whether the complete symmetric digraph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msubsup>\u0000 <mi>K</mi>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>*</mo>\u0000 </msubsup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, assuming <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <msub>\u0000 <mi>m</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>⋯</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <msub>\u0000 <mi>m</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, admits a decomposition into spanning subdigraphs, each a disjoint un","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 7","pages":"239-260"},"PeriodicalIF":0.5,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21967","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143945026","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Putatively Optimal Projective Spherical Designs With Little Apparent Symmetry","authors":"Alex Elzenaar,&nbsp;Shayne Waldron","doi":"10.1002/jcd.21979","DOIUrl":"https://doi.org/10.1002/jcd.21979","url":null,"abstract":"<p>We give some new explicit examples of putatively optimal projective spherical designs, that is, ones for which there is numerical evidence that they are of minimal size. These form continuous families, and so have little apparent symmetry in general, which requires the introduction of new techniques for their construction. New examples of interest include an 11-point spherical <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mn>3</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-design for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>R</mi>\u0000 \u0000 <mn>3</mn>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, and a 12-point spherical <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-design for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>R</mi>\u0000 \u0000 <mn>4</mn>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> given by four Mercedes-Benz frames that lie on equi-isoclinic planes. The latter example shows that the set of optimal spherical designs can be uncountable. We also give results of an extensive numerical study to determine the nature of the real algebraic variety of optimal projective real spherical designs, and in particular when it is a single point (a unique design) or corresponds to an infinite family of designs.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 6","pages":"222-234"},"PeriodicalIF":0.5,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21979","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143845833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Classification of the Flag-Transitive 2-\u0000 \u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 v\u0000 ,\u0000 3\u0000 ,\u0000 λ\u0000 \u0000 )\u0000 \u0000 \u0000 \u0000 Designs With \u0000 \u0000 \u0000 \u0000 v\u0000 ≡\u0000 1\u0000 ,\u0000 3\u0000 \u0000 \u0000 (\u0000 \u0000 mod\u0000 \u0000 6\u0000 \u0000 )\u0000 \u0000 \u0000 \u0000 and \u0000 \u0000 \u0000 \u0000 v\u0000 ≡\u0000 6\u0000 \u0000 \u0000 (\u0000 \u0000 mod\u0000 \u0000 λ\u0000 \u0000 )","authors":"Alessandro Montinaro,&nbsp;Eliana Francot","doi":"10.1002/jcd.21978","DOIUrl":"https://doi.org/10.1002/jcd.21978","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;In this paper, we provide a complete classification of the 2-&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; designs with &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;≡&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;mod&lt;/mi&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 \u0000 &lt;mn&gt;6&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;≡&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;6&lt;/mn&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;mod&lt;/mi&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 \u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; admitting a flag-transitive automorphism group non-isomorphic to a subgroup of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;Γ&lt;/mi&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 6","pages":"217-221"},"PeriodicalIF":0.5,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143846041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"GDD Type Spanning Bipartite Block Designs","authors":"Shoko Chisaki,&nbsp;Ryoh Fuji-Hara,&nbsp;Nobuko Miyamoto","doi":"10.1002/jcd.21976","DOIUrl":"https://doi.org/10.1002/jcd.21976","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;There is a one-to-one correspondence between the point set of a group divisible design (GDD) with &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; groups of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; points and the edge set of a complete bipartite graph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. A block of GDD corresponds to a subgraph of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We show that the concurrence conditions of two points of GDD can correspond to the edge concurrence conditions of subgraphs of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 6","pages":"207-216"},"PeriodicalIF":0.5,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143845859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cycle Switching in Steiner Triple Systems of Order 19","authors":"Grahame Erskine,&nbsp;Terry S. Griggs","doi":"10.1002/jcd.21975","DOIUrl":"https://doi.org/10.1002/jcd.21975","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;Cycle switching is a particular form of transformation applied to isomorphism classes of a Steiner triple system of a given order &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; (an &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mtext&gt;STS&lt;/mtext&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;), yielding another &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mtext&gt;STS&lt;/mtext&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. This relationship may be represented by an undirected graph. An &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mtext&gt;STS&lt;/mtext&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; admits cycles of lengths &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;6&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;…&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;7&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In the particular case of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 5","pages":"195-204"},"PeriodicalIF":0.5,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143595458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}