Journal of Combinatorial Designs最新文献

{"title":"On the existence of \u0000 \u0000 \u0000 k\u0000 \u0000 $k$\u0000 -cycle semiframes for even \u0000 \u0000 \u0000 k\u0000 \u0000 $k$","authors":"Li Wang, Haibo Ji, Haitao Cao","doi":"10.1002/jcd.21908","DOIUrl":"https://doi.org/10.1002/jcd.21908","url":null,"abstract":"<p>A <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${C}_{k}$</annotation>\u0000 </semantics></math>-semiframe of type <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>g</mi>\u0000 <mi>u</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation> ${g}^{u}$</annotation>\u0000 </semantics></math> is a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${C}_{k}$</annotation>\u0000 </semantics></math>-group divisible design of type <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>g</mi>\u0000 <mi>u</mi>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 <mo>,</mo>\u0000 <mi>G</mi>\u0000 <mo>,</mo>\u0000 <mi>ℬ</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${g}^{u}({mathscr{X}},{mathscr{G}},{rm{ {mathcal B} }})$</annotation>\u0000 </semantics></math> in which <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{X}}$</annotation>\u0000 </semantics></math> is the vertex set, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{G}}$</annotation>\u0000 </semantics></math> is the group set, and the set <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ℬ</mi>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal B} }}$</annotation>\u0000 </semantics></math> of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-cycles can be written as a disjoint union <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ℬ</mi>\u0000 <mo>=</mo>\u0000 <mi>P</mi>\u0000 <mo>∪</mo>\u0000 <mi>Q</mi>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal B} }}={mathscr{P}}cup {mathscr{Q}}$</annotation>\u0000 </semantics></math> where <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{P}}$</annotation>\u0000 </semantics></math> is partitioned into parallel classes","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 10","pages":"511-530"},"PeriodicalIF":0.7,"publicationDate":"2023-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50131264","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":"Totally symmetric quasigroups of order 16","authors":"Hy Ginsberg","doi":"10.1002/jcd.21910","DOIUrl":"https://doi.org/10.1002/jcd.21910","url":null,"abstract":"<p>We present the number of totally symmetric quasigroups (equivalently, totally symmetric Latin squares) of order 16, as well as the number of isomorphism classes of such objects. Totally symmetric quasigroups of orders up to and including 16 that are (respectively) medial, idempotent, and unipotent are also enumerated.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 10","pages":"531-542"},"PeriodicalIF":0.7,"publicationDate":"2023-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50131265","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":"Enumerating Steiner triple systems","authors":"Daniel Heinlein,&nbsp;Patric R. J. Östergård","doi":"10.1002/jcd.21906","DOIUrl":"https://doi.org/10.1002/jcd.21906","url":null,"abstract":"<p>Steiner triple systems (STSs) have been classified up to order 19. Earlier estimations of the number of isomorphism classes of STSs of order 21, the smallest open case, are discouraging as for classification, so it is natural to focus on the easier problem of merely counting the isomorphism classes. Computational approaches for counting STSs are here considered and lead to an algorithm that is used to obtain the number of isomorphism classes for order 21: 14,796,207,517,873,771.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 10","pages":"479-495"},"PeriodicalIF":0.7,"publicationDate":"2023-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21906","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50140412","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":"Cycles of quadratic Latin squares and antiperfect 1-factorisations","authors":"Jack Allsop","doi":"10.1002/jcd.21905","DOIUrl":"https://doi.org/10.1002/jcd.21905","url":null,"abstract":"<p>A Latin square of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> is an <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $ntimes n$</annotation>\u0000 </semantics></math> matrix of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> symbols, such that each symbol occurs exactly once in each row and column. For an odd prime power <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 <annotation> $q$</annotation>\u0000 </semantics></math> let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mi>q</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathbb{F}}}_{q}$</annotation>\u0000 </semantics></math> denote the finite field of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 <annotation> $q$</annotation>\u0000 </semantics></math>. A quadratic Latin square is a Latin square <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 \u0000 <mrow>\u0000 <mo>[</mo>\u0000 \u0000 <mrow>\u0000 <mi>a</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>b</mi>\u0000 </mrow>\u0000 \u0000 <mo>]</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal L} }}[a,b]$</annotation>\u0000 </semantics></math> defined by\u0000\u0000 </p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 9","pages":"447-475"},"PeriodicalIF":0.7,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21905","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50127978","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 chromatic index of finite projective spaces","authors":"Lei Xu,&nbsp;Tao Feng","doi":"10.1002/jcd.21904","DOIUrl":"https://doi.org/10.1002/jcd.21904","url":null,"abstract":"<p>A line coloring of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>PG</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{PG}(n,q)$</annotation>\u0000 </semantics></math>, the <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-dimensional projective space over GF<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>q</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $(q)$</annotation>\u0000 </semantics></math>, is an assignment of colors to all lines of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>PG</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{PG}(n,q)$</annotation>\u0000 </semantics></math> so that any two lines with the same color do not intersect. The chromatic index of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>PG</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{PG}(n,q)$</annotation>\u0000 </semantics></math>, denoted by <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>χ</mi>\u0000 \u0000 <mo>′</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mtext>PG</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 9","pages":"432-446"},"PeriodicalIF":0.7,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50148157","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":"Existence of small ordered orthogonal arrays","authors":"Kai-Uwe Schmidt,&nbsp;Charlene Weiß","doi":"10.1002/jcd.21903","DOIUrl":"https://doi.org/10.1002/jcd.21903","url":null,"abstract":"<p>We show that there exist ordered orthogonal arrays, whose sizes deviate from the Rao bound by a factor that is polynomial in the parameters of the ordered orthogonal array. The proof is nonconstructive and based on a probabilistic method due to Kuperberg, Lovett and Peled.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 9","pages":"422-431"},"PeriodicalIF":0.7,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21903","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50152419","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":"Linear and circular single-change covering designs revisited","authors":"Amanda Chafee,&nbsp;Brett Stevens","doi":"10.1002/jcd.21885","DOIUrl":"https://doi.org/10.1002/jcd.21885","url":null,"abstract":"&lt;p&gt;A &lt;i&gt;single-change covering design&lt;/i&gt; (SCCD) is a &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $v$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-set &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $X$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and an ordered list &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ℒ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${rm{ {mathcal L} }}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;b&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $b$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; blocks of size &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; where every pair from &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $X$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single-change covering design, or more simply, a single-change covering design. A single-change covering design is circular when the first and last blocks also differ by one element. A single-change covering design is minimum if no other smaller design can be constructed for a given &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $v,k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In this paper, we use a new recursive construction to solve the existence of circular SCCD(&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;b&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $v,4,b$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;) for all &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $v$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and three residue classes of circular SCCD(&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;5&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;b&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $v,5,b$&lt;/annotation&gt;\u0000 &lt;/semantics","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 9","pages":"405-421"},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21885","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50116119","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":"Self-dual association schemes, fusions of Hamming schemes, and partial geometric designs","authors":"Bangteng Xu","doi":"10.1002/jcd.21889","DOIUrl":"https://doi.org/10.1002/jcd.21889","url":null,"abstract":"&lt;p&gt;Partial geometric designs can be constructed from basic relations of association schemes. An infinite family of partial geometric designs were constructed from the fusion schemes of certain Hamming schemes in work by Nowak et al. (2016). A general method to create partial geometric designs from association schemes is given by Xu (2023). In this paper, we continue the research by Xu (2023). We will first study the properties and characterizations of self-dual association schemes. Then using the characterizations of self-dual association schemes and the representation theory (character tables) of commutative association schemes, we obtain characterizations and classifications of self-dual (symmetric or nonsymmetric) association schemes of rank 4 that produce as many as possible nontrivial partial geometric designs or 2-designs. In particular, for a primitive self-dual symmetric association scheme &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mo&gt;}&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;≤&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;i&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;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${mathscr{X}}=(X,{{{R}_{i}}}_{0le ile 3})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of rank 4, if &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;∣&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;∣&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $| X| $&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a power of 3 and each of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 8","pages":"373-399"},"PeriodicalIF":0.7,"publicationDate":"2023-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50120463","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":"Primitive \u0000 \u0000 \u0000 \u0000 C\u0000 4\u0000 \u0000 \u0000 ${C}_{4}$\u0000 -decompositions of \u0000 \u0000 \u0000 \u0000 K\u0000 n\u0000 \u0000 −\u0000 I\u0000 \u0000 ${K}_{n}-I$","authors":"Michael W. Schroeder","doi":"10.1002/jcd.21887","DOIUrl":"https://doi.org/10.1002/jcd.21887","url":null,"abstract":"<p>A decomposition <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{C}}$</annotation>\u0000 </semantics></math> of a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is primitive if no proper, nontrivial subset of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{C}}$</annotation>\u0000 </semantics></math> is a decomposition of an induced subgraph of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. An unresolved question posed by Asplund et al. in a recent publication involves the existence of primitive decompositions of cocktail party graphs into cycles of length 4, which is resolved by this paper.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 8","pages":"368-372"},"PeriodicalIF":0.7,"publicationDate":"2023-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50122041","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":"Magic partially filled arrays on abelian groups","authors":"Fiorenza Morini,&nbsp;Marco Antonio Pellegrini","doi":"10.1002/jcd.21886","DOIUrl":"https://doi.org/10.1002/jcd.21886","url":null,"abstract":"&lt;p&gt;In this paper we introduce a special class of partially filled arrays. A magic partially filled array &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mtext&gt;MPF&lt;/mtext&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;;&lt;/mo&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${text{MPF}}_{{rm{Omega }}}(m,n;s,k)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; on a subset &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${rm{Omega }}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of an abelian group &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;Γ&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $({rm{Gamma }},+)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a partially filled array of size &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $mtimes n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with entries in &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${rm{Omega }}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; such that (i) every &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ω&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $omega in {rm{Omega }}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; appears once in the array; (ii) each row contains &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $s$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; filled cells and each column contains &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; filled cells; (iii) there exist (not necessarily distinct) elements &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;y&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 8","pages":"347-367"},"PeriodicalIF":0.7,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21886","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50120677","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}