In this article, we study extensions of Steiner triple systems by means of the associated Steiner loops. We recognize that the set of Veblen points of a Steiner triple system corresponds to the center of the Steiner loop. We investigate extensions of Steiner loops, focusing in particular on the case of Schreier extensions, which provide a powerful method for constructing Steiner triple systems containing Veblen points.
{"title":"Extensions of Steiner Triple Systems","authors":"Giovanni Falcone, Agota Figula, Mario Galici","doi":"10.1002/jcd.21964","DOIUrl":"https://doi.org/10.1002/jcd.21964","url":null,"abstract":"<p>In this article, we study extensions of Steiner triple systems by means of the associated Steiner loops. We recognize that the set of Veblen points of a Steiner triple system corresponds to the center of the Steiner loop. We investigate extensions of Steiner loops, focusing in particular on the case of Schreier extensions, which provide a powerful method for constructing Steiner triple systems containing Veblen points.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 3","pages":"94-108"},"PeriodicalIF":0.5,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21964","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143112441","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}
In this article, we study the BM quasi-Hermitian varieties, laying in the three-dimensional Desarguesian projective space of even order. After a brief investigation of their combinatorial properties, we first show that all of these varieties are projectively equivalent, exhibiting a behavior which is strikingly different from what happens in odd characteristic. This completes the classification project started there. Here we prove more; indeed, by using previous results, we explicitly determine the structure of the full collineation group stabilizing these varieties. Finally, as a byproduct of our investigation, we also construct a family of simple orthogonal arrays , with entries in , where is an even prime power. Orthogonal arrays (OA's) are principally used to minimize the number of experiments needed to investigate how variables in testing interact with each other.
{"title":"On Quasi-Hermitian Varieties in Even Characteristic and Related Orthogonal Arrays","authors":"Angela Aguglia, Luca Giuzzi, Alessandro Montinaro, Viola Siconolfi","doi":"10.1002/jcd.21966","DOIUrl":"https://doi.org/10.1002/jcd.21966","url":null,"abstract":"<p>In this article, we study the BM quasi-Hermitian varieties, laying in the three-dimensional Desarguesian projective space of even order. After a brief investigation of their combinatorial properties, we first show that all of these varieties are projectively equivalent, exhibiting a behavior which is strikingly different from what happens in odd characteristic. This completes the classification project started there. Here we prove more; indeed, by using previous results, we explicitly determine the structure of the full collineation group stabilizing these varieties. Finally, as a byproduct of our investigation, we also construct a family of simple orthogonal arrays <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>O</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>5</mn>\u0000 </msup>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>4</mn>\u0000 </msup>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>q</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, with entries in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mi>q</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is an even prime power. Orthogonal arrays (OA's) are principally used to minimize the number of experiments needed to investigate how variables in testing interact with each other.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 3","pages":"109-122"},"PeriodicalIF":0.5,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21966","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143112442","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}
<div> <p>Let <span></span><math> <semantics> <mrow> <mrow> <mi>q</mi> </mrow> </mrow> </semantics></math> be a prime power and <span></span><math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </semantics></math> be a natural number. What are the possible cardinalities of point sets <span></span><math> <semantics> <mrow> <mrow> <mi>S</mi> </mrow> </mrow> </semantics></math> in a projective plane of order <span></span><math> <semantics> <mrow> <mrow> <mi>q</mi> </mrow> </mrow> </semantics></math>, which do not intersect any line at exactly <span></span><math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </semantics></math> points? This problem and its variants have been investigated before, in relation with blocking sets, untouchable sets or sets of even type, among others. In this article, we show a series of results which point out the existence of all or almost all possible values <span></span><math> <semantics> <mrow> <mrow> <mi>m</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>0</mn> <mo>,</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> </mrow> </mrow> </semantics></math> for <span></span><math> <semantics> <mrow> <mrow> <mo>∣</mo> <mi>S</mi> <mo>∣</mo> <mo>=<
{"title":"Avoiding Secants of Given Size in Finite Projective Planes","authors":"Tamás Héger, Zoltán Lóránt Nagy","doi":"10.1002/jcd.21968","DOIUrl":"https://doi.org/10.1002/jcd.21968","url":null,"abstract":"<div>\u0000 \u0000 <p>Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be a prime power and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be a natural number. What are the possible cardinalities of point sets <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> in a projective plane of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, which do not intersect any line at exactly <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> points? This problem and its variants have been investigated before, in relation with blocking sets, untouchable sets or sets of even type, among others. In this article, we show a series of results which point out the existence of all or almost all possible values <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mrow>\u0000 <mo>[</mo>\u0000 \u0000 <mrow>\u0000 <mn>0</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>q</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\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 <mo>∣</mo>\u0000 \u0000 <mi>S</mi>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mo>=<","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 3","pages":"83-93"},"PeriodicalIF":0.5,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143120778","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}
Peter Danziger, Melissa A. Huggan, Rehan Malik, Trent G. Marbach
<p>We consider playing the game of Tic-Tac-Toe on block designs BIBD<span></span><math> <semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $(v,k,lambda )$</annotation> </semantics></math> and transversal designs TD<span></span><math> <semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>,</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $(k,n)$</annotation> </semantics></math>. Players take turns choosing points and the first player to complete a block wins the game. We show that triple systems, BIBD<span></span><math> <semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $(v,3,lambda )$</annotation> </semantics></math>, are a first-player win if and only if <span></span><math> <semantics> <mrow> <mi>v</mi> <mo>≥</mo> <mn>5</mn> </mrow> <annotation> $vge 5$</annotation> </semantics></math>. Further, we show that for <span></span><math> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn>
<div> <p>There are four nonisomorphic configurations of triples that can form a triangle in a three-uniform hypergraph, where the configurations <span></span><math> <semantics> <mrow> <mi>B</mi> </mrow> <annotation> ${bf{B}}$</annotation> </semantics></math> and <span></span><math> <semantics> <mrow> <mi>D</mi> </mrow> <annotation> ${bf{D}}$</annotation> </semantics></math> on <span></span><math> <semantics> <mrow> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>}</mo> </mrow> <annotation> ${1,2,3,4,5}$</annotation> </semantics></math> consist of three triples <span></span><math> <semantics> <mrow> <mn>125</mn> <mo>,</mo> <mn>134</mn> <mo>,</mo> <mn>234</mn> </mrow> <annotation> $125,134,234$</annotation> </semantics></math> and <span></span><math> <semantics> <mrow> <mn>123</mn> <mo>,</mo> <mn>134</mn> <mo>,</mo> <mn>235</mn> </mrow> <annotation> $123,134,235$</annotation> </semantics></math>, respectively. Denote by ex<span></span><math> <semantics> <mrow> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>D</mi> <mo>)</mo> </mrow> <annotation> $(n,{bf{D}})$</annotation> </semantics></math> and ex<span></span><math> <semantics> <mrow> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>BD</mi> <mo>)</mo> </mrow> <annotation> $(n,{bf{BD}})$</annotation> </semantics></math> the maximum number of triples in a three-uniform hypergraph on <span></span><m
Jingyuan Chen, Huangsheng Yu, R. Julian R. Abel, Dianhua Wu
<p>A design is said to be <i>super-simple</i> if the intersection of any two blocks has at most two elements. A design with index <span></span><math> <semantics> <mrow> <mi>λ</mi> </mrow> <annotation> $lambda $</annotation> </semantics></math> is said to be <i>completely reducible</i>, if its blocks can be partitioned into nonempty collections <span></span><math> <semantics> <mrow> <msub> <mi>B</mi> <mi>i</mi> </msub> <mo>,</mo> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>λ</mi> </mrow> <annotation> ${{mathscr{B}}}_{i},1le ile lambda $</annotation> </semantics></math>, such that each <span></span><math> <semantics> <mrow> <msub> <mi>B</mi> <mi>i</mi> </msub> </mrow> <annotation> ${{mathscr{B}}}_{i}$</annotation> </semantics></math> together with the point set forms a design with index unity. In this paper, it is proved that there exists a completely reducible super-simple (CRSS) <span></span><math> <semantics> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <annotation> $(v,4,4)$</annotation> </semantics></math> balanced incomplete block design (<span></span><math> <semantics> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <annotation> $(v,4,4)$</annotation> </semantics></math>-BIBD for short) if and only if <span></span><math> <semantics> <mrow> <mi>v</mi> <mo>≥</mo> <mn>13</mn> </mrow> <annotation> $vge 13$</annotation> </semantics></math>
( v , d , w ) q ${(v,d,w)}_{q}$ 编码的最大大小记为 A q ( v , d , w ) ${A}_{q}(v,d,w)$ ，达到这一大小的 ( v , d , w ) q ${(v,d,w)}_{q}$ 编码称为最优编码。索引为 q - 1 $q-1$ 的 CRSS 设计与 q $q$ -ary CWC 密切相关。利用 CRSS ( v , 4 , 4 ) $(v,4,4)$ -BIBDs 的结果，可以确定 A 5 ( v , 6 , 4 ) ${A}_{5}(v,6,4)$ s 适用于所有 v ≡ 0 , 1 , 3 , 4 ( mod 12 ) , v ≥ 12 $vequiv 0,1,3,4,(mathrm{mod},12),vge 12$ .
{"title":"Completely reducible super-simple \u0000 \u0000 \u0000 (\u0000 \u0000 v\u0000 ,\u0000 4\u0000 ,\u0000 4\u0000 \u0000 )\u0000 \u0000 $(v,4,4)$\u0000 -BIBDs and related constant weight codes","authors":"Jingyuan Chen, Huangsheng Yu, R. Julian R. Abel, Dianhua Wu","doi":"10.1002/jcd.21958","DOIUrl":"https://doi.org/10.1002/jcd.21958","url":null,"abstract":"<p>A design is said to be <i>super-simple</i> if the intersection of any two blocks has at most two elements. A design with index <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation> $lambda $</annotation>\u0000 </semantics></math> is said to be <i>completely reducible</i>, if its blocks can be partitioned into nonempty collections <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>B</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>i</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation> ${{mathscr{B}}}_{i},1le ile lambda $</annotation>\u0000 </semantics></math>, such that each <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>B</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathscr{B}}}_{i}$</annotation>\u0000 </semantics></math> together with the point set forms a design with index unity. In this paper, it is proved that there exists a completely reducible super-simple (CRSS) <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>4</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $(v,4,4)$</annotation>\u0000 </semantics></math> balanced incomplete block design (<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>4</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $(v,4,4)$</annotation>\u0000 </semantics></math>-BIBD for short) if and only if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>13</mn>\u0000 </mrow>\u0000 <annotation> $vge 13$</annotation>\u0000 </semantics></math>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 1","pages":"27-36"},"PeriodicalIF":0.5,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142665937","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}
In this paper, we characterise point sets having the same intersection numbers with respect to hyperplanes as an ovoidal cone. In particular, we show that a set of points of which blocks all planes and intersects solids in , or points is a plane or an ovoidal cone, and determine all examples that arise when the blocking condition is omitted.
{"title":"Characterising ovoidal cones by their hyperplane intersection numbers","authors":"Bart De Bruyn, Geertrui Van de Voorde","doi":"10.1002/jcd.21959","DOIUrl":"https://doi.org/10.1002/jcd.21959","url":null,"abstract":"<p>In this paper, we characterise point sets having the same intersection numbers with respect to hyperplanes as an ovoidal cone. In particular, we show that a set of points of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>PG</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mn>4</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{PG}(4,q)$</annotation>\u0000 </semantics></math> which blocks all planes and intersects solids in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> $q+1$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> ${q}^{2}+1$</annotation>\u0000 </semantics></math> or <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>q</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> ${q}^{2}+q+1$</annotation>\u0000 </semantics></math> points is a plane or an ovoidal cone, and determine all examples that arise when the blocking condition is omitted.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 1","pages":"5-26"},"PeriodicalIF":0.5,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21959","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142665775","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}
An internal or friendly partition of a vertex set of a graph is a partition to two nonempty sets 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.
图 G $G$ 的顶点集 V ( G ) $V(G)$的内部或友好分区是对两个非空集 A ∪ B $Acup B$ 的分区，使得每个顶点在自己的类中至少有和在另一个类中一样多的邻居。受 Diwan 关于高周长图内部分区存在性证明的启发，我们给出了投影平面入射图内部分区存在性的构造性证明，并讨论了其几何性质。此外，我们还精确地确定了对于平方阶的德萨格平面的入射图，所有顶点同时可以达到的本类邻集与他类邻集的最大可能差值。
{"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}
Marco Buratti, Donald L. Kreher, Douglas R. Stinson
<p>In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the <i>nested point</i>) to every block of a <span></span><math> <semantics> <mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> $(v,k,lambda )$</annotation> </semantics></math>-BIBD in such a way that we end up with a partial <span></span><math> <semantics> <mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>λ</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> $(v,k+1,lambda +1)$</annotation> </semantics></math>-BIBD. In the case where the partial <span></span><math> <semantics> <mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>λ</mi>
在平衡不完全区块设计（或 BIBD）的嵌套中，我们希望在 ( v , k , λ ) $(v,k,lambda )$ -BIBD 的每个区块中添加一个点（嵌套点），这样我们最终会得到一个部分 ( v , k + 1 , λ + 1 ) $(v,k+1,lambda +1)$ -BIBD 。在部分 ( v , k + 1 , λ + 1 ) $(v,k+1,lambda +1)$ -BIBD 实际上是 ( v , k + 1 , λ + 1 ) $(v,k+1,lambda +1)$ -BIBD 的情况下，我们有一个完美嵌套。我们证明，当且仅当 k = 2 λ + 1 $k=2lambda +1$ 时，嵌套是完美的。完美嵌套以前已知存在于斯坦纳三重系统中（即
{"title":"Nestings of BIBDs with block size four","authors":"Marco Buratti, Donald L. Kreher, Douglas R. Stinson","doi":"10.1002/jcd.21957","DOIUrl":"https://doi.org/10.1002/jcd.21957","url":null,"abstract":"<p>In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the <i>nested point</i>) to every block of a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>λ</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(v,k,lambda )$</annotation>\u0000 </semantics></math>-BIBD in such a way that we end up with a partial <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>λ</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(v,k+1,lambda +1)$</annotation>\u0000 </semantics></math>-BIBD. In the case where the partial <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>λ</mi>\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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：