In 1991, Shalaby conjectured that any , where or , admits a strong Skolem starter. In 2018, the authors fully described and explicitly constructed the infinite “cardioidal” family of strong Skolem starters. No other infinite family of these combinatorial designs was known to date. Statements regarding the products of starters, proven in this paper give a new way of generating strong or skew Skolem starters of composite orders. This approach extends our previous result by generating new infinite families of these starters that are not cardioidal.
{"title":"On products of strong Skolem starters","authors":"Oleg Ogandzhanyants, Margarita Kondratieva, Nabil Shalaby","doi":"10.1002/jcd.21943","DOIUrl":"10.1002/jcd.21943","url":null,"abstract":"<p>In 1991, Shalaby conjectured that any <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow></math>, where <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>≡</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow></math> or <span></span><math>\u0000 \u0000 <mrow>\u0000 <mn>3</mn>\u0000 <mspace></mspace>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>mod</mi>\u0000 <mspace></mspace>\u0000 \u0000 <mn>8</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>11</mn>\u0000 </mrow></math>, admits a strong Skolem starter. In 2018, the authors fully described and explicitly constructed the infinite “cardioidal” family of strong Skolem starters. No other infinite family of these combinatorial designs was known to date. Statements regarding the products of starters, proven in this paper give a new way of generating strong or skew Skolem starters of composite orders. This approach extends our previous result by generating new infinite families of these starters that are not cardioidal.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 8","pages":"464-487"},"PeriodicalIF":0.7,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942168","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}
Orthogonal array and a large set of orthogonal arrays are important research objects in combinatorial design theory, and they are widely applied to statistics, computer science, coding theory, and cryptography. In this paper, some new series of large sets of orthogonal arrays are given by direct construction, juxtaposition construction, Hadamard construction, finite field construction, and difference matrix construction. Subsequently, many new infinite classes of orthogonal arrays are obtained by using these large sets of orthogonal arrays and Kronecker product.
{"title":"New results on large sets of orthogonal arrays and orthogonal arrays","authors":"Guangzhou Chen, Xiaodong Niu, Jiufeng Shi","doi":"10.1002/jcd.21944","DOIUrl":"10.1002/jcd.21944","url":null,"abstract":"<p>Orthogonal array and a large set of orthogonal arrays are important research objects in combinatorial design theory, and they are widely applied to statistics, computer science, coding theory, and cryptography. In this paper, some new series of large sets of orthogonal arrays are given by direct construction, juxtaposition construction, Hadamard construction, finite field construction, and difference matrix construction. Subsequently, many new infinite classes of orthogonal arrays are obtained by using these large sets of orthogonal arrays and Kronecker product.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 8","pages":"488-515"},"PeriodicalIF":0.7,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930827","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}
<p>For any finite group <span></span><math> <mrow> <mi>G</mi> </mrow></math>, the Terwilliger algebra <span></span><math> <mrow> <mi>T</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow></math> of the group association scheme satisfies the following inclusions: <span></span><math> <mrow> <msub> <mi>T</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>⊆</mo> <mi>T</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>⊆</mo> <mover> <mi>T</mi> <mo>˜</mo> </mover> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow></math>, where <span></span><math> <mrow> <msub> <mi>T</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow></math> is a specific vector space and <span></span><math> <mrow> <mover> <mi>T</mi> <mo>˜</mo> </mover> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow></math> is the centralizer algebra of the permutation representation of <span></span><math> <mrow> <mi>G</mi> </mrow></math> induced by the action of conjugation. The group <span></span><math> <mrow> <mi>G</mi> </mrow></math> is said to be triply transitive if <span></span><math> <mrow> <msub> <mi>T</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>G</mi>
<p>An <span></span><math> <mrow> <mi>ℓ</mi> </mrow></math>-cycle decomposition of <span></span><math> <mrow> <msub> <mi>K</mi> <mi>v</mi> </msub> </mrow></math> is said to be <i>equitably 2-colourable</i> if there is a 2-vertex-colouring of <span></span><math> <mrow> <msub> <mi>K</mi> <mi>v</mi> </msub> </mrow></math> such that each colour is represented (approximately) an equal number of times on each cycle: more precisely, we ask that in each cycle <span></span><math> <mrow> <mi>C</mi> </mrow></math> of the decomposition, each colour appears on <span></span><math> <mrow> <mrow> <mo>⌊</mo> <mrow> <mi>ℓ</mi> <mo>∕</mo> <mn>2</mn> </mrow> <mo>⌋</mo> </mrow> </mrow></math> or <span></span><math> <mrow> <mrow> <mo>⌈</mo> <mrow> <mi>ℓ</mi> <mo>∕</mo> <mn>2</mn> </mrow> <mo>⌉</mo> </mrow> </mrow></math> of the vertices of <span></span><math> <mrow> <mi>C</mi> </mrow></math>. In this paper we study the existence of equitably 2-colourable <span></span><math> <mrow> <mi>ℓ</mi> </mrow></math>-cycle decompositions of <span></span><math> <mrow> <msub> <mi>K</mi> <mi>v</mi> </msub> </mrow></math>, where <span></span><math> <mrow> <mi>ℓ</mi> </mrow></math> is odd, and prove the existence of such a decomposition for <span></span><math> <mrow> <mi>v</mi> <mo>≡</mo> <mn>1</mn> <mo>,</mo> <mi>ℓ</mi> </mrow></math> (mod <span></span><math>
{"title":"On equitably 2-colourable odd cycle decompositions","authors":"Andrea Burgess, Francesca Merola","doi":"10.1002/jcd.21937","DOIUrl":"10.1002/jcd.21937","url":null,"abstract":"<p>An <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 </mrow></math>-cycle decomposition of <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mi>v</mi>\u0000 </msub>\u0000 </mrow></math> is said to be <i>equitably 2-colourable</i> if there is a 2-vertex-colouring of <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mi>v</mi>\u0000 </msub>\u0000 </mrow></math> such that each colour is represented (approximately) an equal number of times on each cycle: more precisely, we ask that in each cycle <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow></math> of the decomposition, each colour appears on <span></span><math>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>⌊</mo>\u0000 \u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 \u0000 <mo>∕</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 \u0000 <mo>⌋</mo>\u0000 </mrow>\u0000 </mrow></math> or <span></span><math>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>⌈</mo>\u0000 \u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 \u0000 <mo>∕</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 \u0000 <mo>⌉</mo>\u0000 </mrow>\u0000 </mrow></math> of the vertices of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow></math>. In this paper we study the existence of equitably 2-colourable <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 </mrow></math>-cycle decompositions of <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mi>v</mi>\u0000 </msub>\u0000 </mrow></math>, where <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 </mrow></math> is odd, and prove the existence of such a decomposition for <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>≡</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>ℓ</mi>\u0000 </mrow></math> (mod <span></span><math>\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 8","pages":"419-437"},"PeriodicalIF":0.7,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21937","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140630988","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}
<p>Let <span></span><math> <semantics> <mrow> <mi>Γ</mi> </mrow> <annotation> ${rm{Gamma }}$</annotation> </semantics></math> be the graph whose vertices are the chambers of the finite projective 3-space <span></span><math> <semantics> <mrow> <mtext>PG</mtext> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $text{PG}(3,q)$</annotation> </semantics></math>, with two vertices being adjacent if and only if the corresponding chambers are in general position. We show that a maximal independent set of vertices of <span></span><math> <semantics> <mrow> <mi>Γ</mi> </mrow> <annotation> ${rm{Gamma }}$</annotation> </semantics></math> contains <span></span><math> <semantics> <mrow> <msup> <mi>q</mi> <mn>4</mn> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>q</mi> <mn>3</mn> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>3</mn> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> <annotation> ${q}^{4}+3{q}^{3}+4{q}^{2}+3q+1$</annotation> </semantics></math>, or <span></span><math> <semantics> <mrow> <mn>3</mn> <msup> <mi>q</mi> <mn>3</mn> </msup> <mo>+</mo> <mn>5</mn> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>3</mn> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> <annotation> $3{q}^{3}+5{q}^{2}+3q+1$</annotation> </semantics></math>, or at most <span></span><math> <semantics> <mrow> <mn>3</mn> <msup> <mi>
{"title":"Maximal cocliques and the chromatic number of the Kneser graph on chambers of PG\u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 3\u0000 ,\u0000 q\u0000 \u0000 )\u0000 \u0000 \u0000 $(3,q)$","authors":"Philipp Heering, Klaus Metsch","doi":"10.1002/jcd.21940","DOIUrl":"10.1002/jcd.21940","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 </mrow>\u0000 <annotation> ${rm{Gamma }}$</annotation>\u0000 </semantics></math> be the graph whose vertices are the chambers of the finite projective 3-space <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>PG</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <mo>,</mo>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{PG}(3,q)$</annotation>\u0000 </semantics></math>, with two vertices being adjacent if and only if the corresponding chambers are in general position. We show that a maximal independent set of vertices of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 </mrow>\u0000 <annotation> ${rm{Gamma }}$</annotation>\u0000 </semantics></math> contains <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 <mn>4</mn>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <mn>3</mn>\u0000 <msup>\u0000 <mi>q</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <mn>4</mn>\u0000 <msup>\u0000 <mi>q</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <mn>3</mn>\u0000 <mi>q</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> ${q}^{4}+3{q}^{3}+4{q}^{2}+3q+1$</annotation>\u0000 </semantics></math>, or <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <msup>\u0000 <mi>q</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <mn>5</mn>\u0000 <msup>\u0000 <mi>q</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <mn>3</mn>\u0000 <mi>q</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> $3{q}^{3}+5{q}^{2}+3q+1$</annotation>\u0000 </semantics></math>, or at most <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <msup>\u0000 <mi>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 7","pages":"388-409"},"PeriodicalIF":0.7,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21940","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140570787","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}
John Polhill, James A. Davis, Ken W. Smith, Eric Swartz
Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.
{"title":"Genuinely nonabelian partial difference sets","authors":"John Polhill, James A. Davis, Ken W. Smith, Eric Swartz","doi":"10.1002/jcd.21938","DOIUrl":"10.1002/jcd.21938","url":null,"abstract":"<p>Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of <i>genuinely nonabelian</i> PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 7","pages":"351-370"},"PeriodicalIF":0.7,"publicationDate":"2024-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21938","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140570623","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}
R. Julian R. Abel, Thomas Britz, Yudhistira A. Bunjamin, Diana Combe
In this paper, we consider the existence of group divisible designs (GDDs) with block size and group sizes and . We show that there exists a -GDD of type for all but a finite specified set of feasible values for .
在本文中，我们考虑了块大小为 4 $4$、组大小为 4 $4$ 和 7 $7$ 的组可分割设计（GDD）的存在性。我们证明，除了 ( t , s ) $(t,s)$的可行值的有限指定集合外，存在一个 4 $4$ 类型为 4 t 7 s ${4}^{t}{7}^{s}$ 的 4 $4$ -GDD 。
{"title":"Group divisible designs with block size 4 and group sizes 4 and 7","authors":"R. Julian R. Abel, Thomas Britz, Yudhistira A. Bunjamin, Diana Combe","doi":"10.1002/jcd.21932","DOIUrl":"https://doi.org/10.1002/jcd.21932","url":null,"abstract":"<p>In this paper, we consider the existence of group divisible designs (GDDs) with block size <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation> $4$</annotation>\u0000 </semantics></math> and group sizes <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation> $4$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>7</mn>\u0000 </mrow>\u0000 <annotation> $7$</annotation>\u0000 </semantics></math>. We show that there exists a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation> $4$</annotation>\u0000 </semantics></math>-GDD of type <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mn>4</mn>\u0000 <mi>t</mi>\u0000 </msup>\u0000 <msup>\u0000 <mn>7</mn>\u0000 <mi>s</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation> ${4}^{t}{7}^{s}$</annotation>\u0000 </semantics></math> for all but a finite specified set of feasible values for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mo>,</mo>\u0000 <mi>s</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(t,s)$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 6","pages":"328-346"},"PeriodicalIF":0.7,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21932","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140553167","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}
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