Journal of Combinatorial Designs最新文献

{"title":"Partitioning the projective plane into two incidence-rich parts","authors":"Zoltán Lóránt Nagy","doi":"10.1002/jcd.21956","DOIUrl":"https://doi.org/10.1002/jcd.21956","url":null,"abstract":"<p>An internal or friendly partition of a vertex set <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>V</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $V(G)$</annotation>\u0000 </semantics></math> of a graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a partition to two nonempty sets <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>A</mi>\u0000 \u0000 <mo>∪</mo>\u0000 \u0000 <mi>B</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $Acup B$</annotation>\u0000 </semantics></math> such that every vertex has at least as many neighbours in its own class as in the other one. Motivated by Diwan's existence proof on internal partitions of graphs with high girth, we give constructive proofs for the existence of internal partitions in the incidence graph of projective planes and discuss its geometric properties. In addition, we determine exactly the maximum possible difference between the sizes of the neighbour set in its own class and the neighbour set of the other class that can be attained for all vertices at the same time for the incidence graphs of Desarguesian planes of square order.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 12","pages":"703-714"},"PeriodicalIF":0.5,"publicationDate":"2024-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21956","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142429303","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":"Nestings of BIBDs with block size four","authors":"Marco Buratti,&nbsp;Donald L. Kreher,&nbsp;Douglas R. Stinson","doi":"10.1002/jcd.21957","DOIUrl":"https://doi.org/10.1002/jcd.21957","url":null,"abstract":"&lt;p&gt;In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the &lt;i&gt;nested point&lt;/i&gt;) to every block of a &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;mi&gt;k&lt;/mi&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;annotation&gt; $(v,k,lambda )$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-BIBD in such a way that we end up with a partial &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;mi&gt;k&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;mi&gt;λ&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;1&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;annotation&gt; $(v,k+1,lambda +1)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-BIBD. In the case where the partial &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;mi&gt;k&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;mi&gt;λ&lt;/mi&gt;\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 12","pages":"715-743"},"PeriodicalIF":0.5,"publicationDate":"2024-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21957","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142429304","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":"Every latin hypercube of order 5 has transversals","authors":"Alexey L. Perezhogin,&nbsp;Vladimir N. Potapov,&nbsp;Sergey Yu. Vladimirov","doi":"10.1002/jcd.21954","DOIUrl":"10.1002/jcd.21954","url":null,"abstract":"&lt;p&gt;We prove that for all &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;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $ngt 1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; every latin &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;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-dimensional cube of order 5 has transversals. We find all 123 paratopy classes of layer-latin cubes of order 5 with no transversals. For each &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;n&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;annotation&gt; $nge 3$&lt;/annotation&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;q&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;annotation&gt; $qge 3$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; we construct a &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;mn&gt;2&lt;/mn&gt;\u0000 \u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;q&lt;/mi&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;q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $(2q-2)times qtimes {rm{cdots }}times q$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; latin &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;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-dimensional cuboid of order &lt;span&gt;&lt;/span&gt;&lt;ma","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 11","pages":"679-699"},"PeriodicalIF":0.5,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863809","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":"Generalised evasive subspaces","authors":"Anina Gruica,&nbsp;Alberto Ravagnani,&nbsp;John Sheekey,&nbsp;Ferdinando Zullo","doi":"10.1002/jcd.21953","DOIUrl":"10.1002/jcd.21953","url":null,"abstract":"<p>We introduce and explore a new concept of evasive subspace with respect to a collection of subspaces sharing a common dimension, most notably partial spreads. We show that this concept generalises known notions of subspace scatteredness and evasiveness. We establish various upper bounds for the dimension of an evasive subspace with respect to arbitrary partial spreads, obtaining improvements for the Desarguesian ones. We also establish existence results for evasive spaces in a nonconstructive way, using a graph theory approach. The upper and lower bounds we derive have a precise interpretation as bounds for the critical exponent of certain combinatorial geometries. Finally, we investigate connections between the notion of evasive space we introduce and the theory of rank-metric codes, obtaining new results on the covering radius and on the existence of minimal vector rank-metric codes.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 11","pages":"642-678"},"PeriodicalIF":0.5,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141574803","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 eigenfunctions of the block graphs of geometric Steiner systems","authors":"Sergey Goryainov,&nbsp;Dmitry Panasenko","doi":"10.1002/jcd.21951","DOIUrl":"10.1002/jcd.21951","url":null,"abstract":"<p>This paper lies in the context of the studies of eigenfunctions of graphs having minimum cardinality of support. One of the tools is the weight-distribution bound, a lower bound on the cardinality of support of an eigenfunction of a distance-regular graph corresponding to a nonprincipal eigenvalue. The tightness of the weight-distribution bound was previously shown in general for the smallest eigenvalue of a Grassmann graph. However, a characterisation of optimal eigenfunctions was not obtained. Motivated by this open problem, we consider the class of strongly regular Grassmann graphs and give the required characterisation in this case. We then show the tightness of the weight-distribution bound for block graphs of affine designs (defined on the lines of an affine space with two lines being adjacent when intersect) and obtain a similar characterisation of optimal eigenfunctions.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 11","pages":"629-641"},"PeriodicalIF":0.5,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141502912","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":"Symmetric 2-\u0000 \u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 36\u0000 ,\u0000 15\u0000 ,\u0000 6\u0000 \u0000 )\u0000 \u0000 \u0000 \u0000 $(36,15,6)$\u0000 designs with an automorphism of order two","authors":"Sanja Rukavina,&nbsp;Vladimir D. Tonchev","doi":"10.1002/jcd.21952","DOIUrl":"https://doi.org/10.1002/jcd.21952","url":null,"abstract":"&lt;p&gt;Bouyukliev, Fack and Winne classified all 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;mn&gt;36&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;15&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&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;annotation&gt; $(36,15,6)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; designs that admit an automorphism of odd prime order, and gave a partial classification of such designs that admit an automorphism of order 2. In this paper, we give the classification of all symmetric 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;mn&gt;36&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;15&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&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;annotation&gt; $(36,15,6)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; designs that admit an automorphism of order two. It is shown that there are exactly &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;1&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;547&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;701&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $1,547,701$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; nonisomorphic such designs, &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;135&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;779&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $135,779$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of which are self-dual designs. The ternary linear codes spanned by the incidence matrices of these designs are computed. Among these codes, there are near-extremal self-dual cod","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 10","pages":"606-624"},"PeriodicalIF":0.5,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967981","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":"Mutual incidence matrix of two balanced incomplete block designs","authors":"Alexander Shramchenko,&nbsp;Vasilisa Shramchenko","doi":"10.1002/jcd.21949","DOIUrl":"https://doi.org/10.1002/jcd.21949","url":null,"abstract":"<p>We propose to consider a mutual incidence matrix <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>M</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $M$</annotation>\u0000 </semantics></math> of two balanced incomplete block designs built on the same finite set. In the simplest case, this matrix reduces to the standard incidence matrix of one block design. We find all eigenvalues of the matrices <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>M</mi>\u0000 \u0000 <msup>\u0000 <mi>M</mi>\u0000 \u0000 <mi>T</mi>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $M{M}^{T}$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>M</mi>\u0000 \u0000 <mi>T</mi>\u0000 </msup>\u0000 \u0000 <mi>M</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${M}^{T}M$</annotation>\u0000 </semantics></math> and their eigenspaces.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 10","pages":"579-590"},"PeriodicalIF":0.5,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21949","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967979","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":"Infinite series of 3-designs in the extended quadratic residue code","authors":"Madoka Awada","doi":"10.1002/jcd.21950","DOIUrl":"https://doi.org/10.1002/jcd.21950","url":null,"abstract":"<p>In this paper, we show an infinite series of 3-designs in the extended quadratic residue codes over <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <msup>\u0000 <mi>r</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${{mathbb{F}}}_{{r}^{2}}$</annotation>\u0000 </semantics></math> for a prime <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $r$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 10","pages":"591-605"},"PeriodicalIF":0.5,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967980","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":"Zarankiewicz numbers near the triple system threshold","authors":"Guangzhou Chen,&nbsp;Daniel Horsley,&nbsp;Adam Mammoliti","doi":"10.1002/jcd.21948","DOIUrl":"https://doi.org/10.1002/jcd.21948","url":null,"abstract":"&lt;p&gt;For positive integers &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $m$&lt;/annotation&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 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, the Zarankiewicz number &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&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;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${Z}_{2,2}(m,n)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; can be defined as the maximum total degree of a linear hypergraph with &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $m$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; vertices and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; edges. Guy determined &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&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;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${Z}_{2,2}(m,n)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for all &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;⩾&lt;/mo&gt;\u0000 &lt;mfenced&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;/mfenced&gt;\u0000 &lt;mo&gt;∕&lt;/mo&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 9","pages":"556-576"},"PeriodicalIF":0.5,"publicationDate":"2024-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21948","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141597005","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 Oberwolfach problem with loving couples","authors":"Gloria Rinaldi","doi":"10.1002/jcd.21946","DOIUrl":"10.1002/jcd.21946","url":null,"abstract":"<p>We generalize the well-known Oberwolfach problem posed by Ringel in 1967. We suppose to have <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mfrac>\u0000 <mi>v</mi>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mrow>\u0000 <annotation> $frac{v}{2}$</annotation>\u0000 </semantics></math> couples (here <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation> $vge 4$</annotation>\u0000 </semantics></math> is an even integer) and suppose that they have to be seated for several nights at <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math> round tables in such a way that each person seats next to his partner exactly <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation> $rge 0$</annotation>\u0000 </semantics></math> times and next to every other person exactly once. We call this problem the Oberwolfach problem with loving couples. When <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation> $r=0$</annotation>\u0000 </semantics></math>, the problem coincides with the so-called spouse-avoiding variant, which was introduced by Huang, Kotzig, and Rosa in 1979. While if either <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $r=2$</annotation>\u0000 </semantics></math> or <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <annotation> $r$</annotation>\u0000 </semantics></math> equals the number of nights, it corresponds to the spouse-loving variant or to the Honeymoon variant, which was recently studied by Bolohan et al. and by Lepine and Sajna, respectively. In this paper, for each possible choice of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <annotation> $r$</annotation>\u0000 </semantics></math>, we construct many classes of solutions to the Oberwolfach problem with loving couples. We also obtain new solutions to the Honeymoon variant.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 9","pages":"532-545"},"PeriodicalIF":0.5,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141120940","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}