Journal of Combinatorial Designs最新文献_第3页

Mutual incidence matrix of two balanced incomplete block designs 两个平衡不完全区块设计的互现矩阵

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2024-06-17 DOI: 10.1002/jcd.21949

Alexander Shramchenko, Vasilisa Shramchenko

{"title":"Mutual incidence matrix of two balanced incomplete block designs","authors":"Alexander Shramchenko, Vasilisa Shramchenko","doi":"10.1002/jcd.21949","DOIUrl":"https://doi.org/10.1002/jcd.21949","url":null,"abstract":"We propose to consider a mutual incidence matrix <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 <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 <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.","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}

引用次数: 0

Infinite series of 3-designs in the extended quadratic residue code 扩展二次残差码中的 3-设计无限序列

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2024-06-17 DOI: 10.1002/jcd.21950

Madoka Awada

引用次数: 0

Zarankiewicz numbers near the triple system threshold 接近三重系统临界值的扎兰凯维奇数

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2024-06-02 DOI: 10.1002/jcd.21948

Guangzhou Chen, Daniel Horsley, Adam Mammoliti

{"title":"Zarankiewicz numbers near the triple system threshold","authors":"Guangzhou Chen, Daniel Horsley, Adam Mammoliti","doi":"10.1002/jcd.21948","DOIUrl":"https://doi.org/10.1002/jcd.21948","url":null,"abstract":"For positive integers <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>, the Zarankiewicz number <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 <mo>,</mo>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${Z}_{2,2}(m,n)$</annotation>\u0000 </semantics></math> can be defined as the maximum total degree of a linear hypergraph with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math> vertices and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> edges. Guy determined <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 <mo>,</mo>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${Z}_{2,2}(m,n)$</annotation>\u0000 </semantics></math> for all <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>⩾</mo>\u0000 <mfenced>\u0000 <mfrac>\u0000 <mi>m</mi>\u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mfenced>\u0000 <mo>∕</mo>\u0000 <mn>3</mn>\u0000 <mo>+</mo>\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}

引用次数: 0

The Oberwolfach problem with loving couples 奥伯沃尔法赫的恩爱夫妻问题

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2024-05-20 DOI: 10.1002/jcd.21946

Gloria Rinaldi

{"title":"The Oberwolfach problem with loving couples","authors":"Gloria Rinaldi","doi":"10.1002/jcd.21946","DOIUrl":"10.1002/jcd.21946","url":null,"abstract":"We generalize the well-known Oberwolfach problem posed by Ringel in 1967. We suppose to have <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 <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 <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 <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 <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 <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 <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 <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.","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}

引用次数: 0

Multifold 1-perfect codes 多倍 1-完美代码

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2024-05-20 DOI: 10.1002/jcd.21947

Denis S. Krotov

{"title":"Multifold 1-perfect codes","authors":"Denis S. Krotov","doi":"10.1002/jcd.21947","DOIUrl":"10.1002/jcd.21947","url":null,"abstract":"A multifold 1-perfect code (1-perfect code for list decoding) in any graph is a set <math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 \u0000 <mi>C</mi>\u0000 </mrow>\u0000 \u0000 <annotation>\u0000 $C$\u0000</annotation>\u0000 </semantics>\u0000 </math> of vertices such that every vertex of the graph is at distance not more than 1 from exactly <math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 \u0000 <mi>μ</mi>\u0000 </mrow>\u0000 \u0000 <annotation>\u0000 $mu $\u0000</annotation>\u0000 </semantics>\u0000 </math> elements of <math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 \u0000 <mi>C</mi>\u0000 </mrow>\u0000 \u0000 <annotation>\u0000 $C$\u0000</annotation>\u0000 </semantics>\u0000 </math>. In <math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 \u0000 <annotation>\u0000 $q$\u0000</annotation>\u0000 </semantics>\u0000 </math>-ary Hamming graphs, where <math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 \u0000 <annotation>\u0000 $q$\u0000</annotation>\u0000 </semantics>\u0000 </math> is a prime power, we characterize all parameters of multifold 1-perfect codes and all parameters of additive multifold 1-perfect codes. In particular, we show that additive multifold 1-perfect codes are related to special multiset generalizations of spreads, multispreads, and that multispreads of parameters corresponding to multifold 1-perfect codes always exist.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 9","pages":"546-555"},"PeriodicalIF":0.5,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141147910","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}

引用次数: 0

Block-transitive triple systems with sporadic or alternating socle 具有零星或交替楔形体的积木式三重体系

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2024-05-20 DOI: 10.1002/jcd.21945

Suyun Ding, Yilin Zhang, Xiaoqin Zhan, Guangzu Chen

{"title":"Block-transitive triple systems with sporadic or alternating socle","authors":"Suyun Ding, Yilin Zhang, Xiaoqin Zhan, Guangzu Chen","doi":"10.1002/jcd.21945","DOIUrl":"10.1002/jcd.21945","url":null,"abstract":"This paper is a contribution to the classification of all pairs <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>G</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $({mathscr{T}},G)$</annotation>\u0000 </semantics></math>, where <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{T}}$</annotation>\u0000 </semantics></math> is a triple system and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a block-transitive but not flag-transitive automorphism group of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{T}}$</annotation>\u0000 </semantics></math>. We prove that if the socle of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a sporadic or alternating group, then one of the following holds:\u0000\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 9","pages":"521-531"},"PeriodicalIF":0.5,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141121084","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}

引用次数: 0

On products of strong Skolem starters 关于强 Skolem 启动器的产物

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2024-05-07 DOI: 10.1002/jcd.21943

Oleg Ogandzhanyants, Margarita Kondratieva, Nabil Shalaby

{"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":"In 1991, Shalaby conjectured that any <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow></math>, where <math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>≡</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow></math> or <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.","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}

引用次数: 0

New results on large sets of orthogonal arrays and orthogonal arrays 关于大型正交阵列集和正交阵列的新成果

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2024-05-07 DOI: 10.1002/jcd.21944

Guangzhou Chen, Xiaodong Niu, Jiufeng Shi

引用次数: 0

The Terwilliger algebras of the group association schemes of three metacyclic groups 三个元环群的群联方案的特尔维利格代数

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2024-04-25 DOI: 10.1002/jcd.21941

Jing Yang, Xiaoqian Zhang, Lihua Feng

{"title":"The Terwilliger algebras of the group association schemes of three metacyclic groups","authors":"Jing Yang, Xiaoqian Zhang, Lihua Feng","doi":"10.1002/jcd.21941","DOIUrl":"10.1002/jcd.21941","url":null,"abstract":"For any finite group <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>, the Terwilliger algebra <math>\u0000 \u0000 <mrow>\u0000 <mi>T</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> of the group association scheme satisfies the following inclusions: <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>T</mi>\u0000 \u0000 <mn>0</mn>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>⊆</mo>\u0000 \u0000 <mi>T</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>⊆</mo>\u0000 \u0000 <mover>\u0000 <mi>T</mi>\u0000 \u0000 <mo>˜</mo>\u0000 </mover>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math>, where <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>T</mi>\u0000 \u0000 <mn>0</mn>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> is a specific vector space and <math>\u0000 \u0000 <mrow>\u0000 <mover>\u0000 <mi>T</mi>\u0000 \u0000 <mo>˜</mo>\u0000 </mover>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> is the centralizer algebra of the permutation representation of <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> induced by the action of conjugation. The group <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is said to be triply transitive if <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>T</mi>\u0000 \u0000 <mn>0</mn>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 8","pages":"438-463"},"PeriodicalIF":0.7,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799847","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}

引用次数: 0

On equitably 2-colourable odd cycle decompositions 关于等效 2 奇循环分解

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2024-04-18 DOI: 10.1002/jcd.21937

Andrea Burgess, Francesca Merola

{"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":"An <math>\u0000 \u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 </mrow></math>-cycle decomposition of <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mi>v</mi>\u0000 </msub>\u0000 </mrow></math> is said to be equitably 2-colourable if there is a 2-vertex-colouring of <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 <math>\u0000 \u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow></math> of the decomposition, each colour appears on <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 <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 <math>\u0000 \u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow></math>. In this paper we study the existence of equitably 2-colourable <math>\u0000 \u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 </mrow></math>-cycle decompositions of <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mi>v</mi>\u0000 </msub>\u0000 </mrow></math>, where <math>\u0000 \u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 </mrow></math> is odd, and prove the existence of such a decomposition for <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 <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}

引用次数: 0