Journal of Combinatorial Designs最新文献_第7页

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 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2024-04-25 DOI: 10.1002/jcd.21941

Jing Yang, Xiaoqian Zhang, Lihua Feng

For any finite group $� � � G �$ , the Terwilliger algebra $� � � T � � � (� � G � �) � �$ of the group association scheme satisfies the following inclusions: $� � � �_{T � � 0 �} � � � (� � G � �) � � � \subseteq � � T � � � (� � G � �) � � � \subseteq � � \overset{T}{�} � � � (� � G � �) � �$ , where $� � � �_{T � � 0 �} � � � (� � G � �) � �$ is a specific vector space and $� � � \overset{T}{�} � � � (� � G � �) � �$ is the centralizer algebra of the permutation representation of $� � � G �$ induced by the action of conjugation. The group $� � � G �$ is said to be triply transitive if $� � � �_{T � � 0 �} � � � (� � G �$

对于任何有限群，群关联方案的特尔维利格代数满足以下结论：其中，是一个特定的向量空间，是共轭作用诱导的的置换表示的中心化代数。如果 .在本文中，我们确定了、和的维数，并证明和是三传递的。此外，我们还给出了、和的特威里格布尔的韦德伯恩成分的完整特征。

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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 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2024-04-18 DOI: 10.1002/jcd.21937

Andrea Burgess, Francesca Merola

An <math> <mrow> <mi>ℓ</mi> </mrow></math>-cycle decomposition of <math> <mrow> <msub> <mi>K</mi> <mi>v</mi> </msub> </mrow></math> is said to be equitably 2-colourable if there is a 2-vertex-colouring of <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 <math> <mrow> <mi>C</mi> </mrow></math> of the decomposition, each colour appears on <math> <mrow> <mrow> <mo>⌊</mo> <mrow> <mi>ℓ</mi> <mo>∕</mo> <mn>2</mn> </mrow> <mo>⌋</mo> </mrow> </mrow></math> or <math> <mrow> <mrow> <mo>⌈</mo> <mrow> <mi>ℓ</mi> <mo>∕</mo> <mn>2</mn> </mrow> <mo>⌉</mo> </mrow> </mrow></math> of the vertices of <math> <mrow> <mi>C</mi> </mrow></math>. In this paper we study the existence of equitably 2-colourable <math> <mrow> <mi>ℓ</mi> </mrow></math>-cycle decompositions of <math> <mrow> <msub> <mi>K</mi> <mi>v</mi> </msub> </mrow></math>, where <math> <mrow> <mi>ℓ</mi> </mrow></math> is odd, and prove the existence of such a decomposition for <math> <mrow> <mi>v</mi> <mo>≡</mo> <mn>1</mn> <mo>,</mo> <mi>ℓ</mi> </mrow></math> (mod <math>

如果存在一种 2 顶点着色，使得每种颜色在每个循环中出现的次数（近似）相等，那么我们就称这种循环分解为等效 2 顶点着色：更确切地说，我们要求在分解的每个循环中，每种颜色都出现在.的顶点或.的顶点上。本文将研究等效 2 顶点着色的循环分解的存在性，其中奇数为(mod )，并证明这种分解的存在性。

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引用次数: 0

Maximal cocliques and the chromatic number of the Kneser graph on chambers of PG ( 3 , q ) $(3,q)$ PG(3,q)室上克奈瑟图的最大茧和色度数

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2024-04-15 DOI: 10.1002/jcd.21940

Philipp Heering, Klaus Metsch

Let <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 <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 <math> <semantics> <mrow> <mi>Γ</mi> </mrow> <annotation> ${rm{Gamma }}$</annotation> </semantics></math> contains <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 <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 <math> <semantics> <mrow> <mn>3</mn> <msup> <mi>

设 Γ${rm{Gamma }}$ 是图，其顶点是有限投影 3 空间 PG(3,q)$text{PG}(3,q)$ 的室，当且仅当相应的室处于一般位置时，两个顶点相邻。我们证明Γ${rm{Gamma }}$的最大独立顶点集包含 q4+3q3+4q2+3q+1${q}^{4}+3{q}^{3}+4{q}^{2}+3q+1${q}^{4}+3{q}^{3}+4{q}^{2}+3q+1$、或 3q3+5q2+3q+1$3{q}^{3}+5{q}^{2}+3q+1$ 或至多 3q3+4q2+3q+2$3{q}^{3}+4{q}^{2}+3q+2$ 元素。对于 q≥4$qge 4$，描述了最大独立集的结构。对于 q≥7$qge 7$，描述了三个最大心数的最大独立集的结构。利用第二大最大独立集的心度，我们证明了Γ${rm{ganma }}$的色度数是 q2+q${q}^{2}+q$.

{"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":"Let <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 <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 <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 </mrow>\u0000 <annotation> ${rm{Gamma }}$</annotation>\u0000 </semantics></math> contains <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 <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 <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}

引用次数: 0

An explicit construction for large sets of infinite dimensional q $q$ -Steiner systems 无穷维 q-Steiner 系统大集合的显式构造

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2024-04-15 DOI: 10.1002/jcd.21942

Daniel R. Hawtin

Let

�            � � � V � � �

be a vector space over the finite field

�            � � � �_{F �                     � q �} � � �

. A

�            � � � q � � �

-Steiner system, or an

�            � � � S �                  � �_{� (�                        � � t �                           �, �                           � k �                           �, �                           � V � �                        �) � �                     � q �} � � �

, is a collection

�            � � � ℬ � � �

of

�            � � � k � � �

-dimensional subspaces of

�            � � � V � � �

such that every

�            � � � t � � �

-dimensional subspace of

�            � � � V � � �

is contained in a unique element of

�            � � � ℬ �

设 V$V$ 是有限域 Fq${{mathbb{F}}}_{q}$ 上的向量空间。一个 q$q$-Steiner 系统或一个 S(t,k,V)q$S{(t,k,V)}_{q}$ 是ℬ${rm{ {mathcal B}}$ 的集合。V$V$ 的 k$k$ 维子空间的集合，使得 V$V$ 的每个 t$t$ 维子空间都包含在ℬ${rm{ {mathcal B}}$ 的唯一元素中。｝}}$.一大组 q$q$-Steiner 系统或 LS(t,k,V)q$LS{(t,k,V)}_{q}$ 是将 V$V$ 的 k$k$ 维子空间划分为 S(t,k,V)q$S{(t,k,V)}_{q}$ 系统。在 V$V$ 具有无限维的情况下，卡梅伦在 1995 年抽象地证明了对于所有有限的 t,kt,k$，1<t<k$1lt tlt k$，LS(t,k,V)q$LS{(t,k,V)}_{q}$的存在。本文明确地构造了 LS(t,t+1,V)q$LS{(t,t+1,V)}_{q}$ ，适用于所有素数幂 q$q$，所有正整数 t$t$，且 V$V$ 具有可数无限维。

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引用次数: 0

Genuinely nonabelian partial difference sets 真正非阿贝尔局部差集

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2024-03-30 DOI: 10.1002/jcd.21938

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.

强规则图（SRGs）为代数组合学提供了一个富饶的探索领域，它综合了图论、线性代数、群论、有限域、有限几何和数论中的技术。尤其令人感兴趣的是那些具有大自形群的 SRG。如果一个自形群有规律地（急剧地）作用于图的顶点，那么我们就可以将图与群的一个子集--部分差集（PDS）--识别开来，这样我们就可以应用群论的技术来研究图。在过去的四十年中，大部分研究工作都集中在利用强大的character theory（特征理论）技术研究无性偏差集。然而，关于非阿贝尔 PDS 的研究却很少。在本文中，我们指出了真正非阿贝尔 PDS 的存在，即参数集的 PDS，其中非阿贝尔群是唯一可能的正则自变群。我们还提出了一些方法，用以证明对于特定参数集或特定 SRG，不可能存在非阿贝尔 PDS。我们描述了四个真正非阿贝尔 PDS 的无限族，其中两个是新的，一个产生于三角形图，一个产生于 Godsil 构建的完整图的 Krein 盖。我们还介绍了通过计算机搜索发现的一种新的非标注 PDS，并提出了一些未来可能的研究方向。

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引用次数: 0

Doubly sequenceable groups 双序列群

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2024-03-30 DOI: 10.1002/jcd.21939

Mohammad Javaheri

Given a sequence <math> <semantics> <mrow> <mi>g</mi> <mo>:</mo> <msub> <mi>g</mi> <mn>0</mn> </msub> <mo>,</mo> <mi>…</mi> <mo>,</mo> <msub> <mi>g</mi> <mi>m</mi> </msub> </mrow> <annotation> ${bf{g}}:{g}_{0},{rm{ldots }},{g}_{m}$</annotation> </semantics></math> in a finite group <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math> with <math> <semantics> <mrow> <msub> <mi>g</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mn>1</mn> <mi>G</mi> </msub> </mrow> <annotation> ${g}_{0}={1}_{G}$</annotation> </semantics></math>, let <math> <semantics> <mrow> <mover> <mi>g</mi> <mo>¯</mo> </mover> <mo>:</mo> <msub> <mover> <mi>g</mi> <mo>¯</mo> </mover> <mn>0</mn> </msub> <mo>,</mo> <mi>…</mi> <mo>,</mo> <msub> <mover> <mi>g</mi> <mo>¯</mo> </mover> <mi>m</mi> </msub> </mrow> <annotation> $bar{{bf{g}}}:{bar{g}}_{0},{rm{ldots }},{bar{g}}_{m}$</annotation> </semantics></math> be the sequence of consecutive quotients of <math> <semantics> <mrow> <mi>g</mi> </mrow> <annotation> ${bf{g}}$</annotation> </semantics></math> defined by <math> <semantics> <mrow> <msub> <mover>

给定有限群 G$G$ 中的序列 g:g0,...,gm${bf{g}}:{g}_{0},{rm{ldots }},{g}_{m}$，其中 g0=1G${g}_{0}={1}_{G}$, 让 g¯:g¯0,...,g¯m$bar{bf{g}}}：{bar{g}}_{0},{rm{ldots }}、{bar{g}}_{m}$ 是 g${bf{g}}$ 的连续商序列，定义为 g¯0=1G${bar{g}}_{0}={1}_{G}$ 和 g¯i=gi-1-1gi${bar{g}}_{i}={g}_{i-1}^{-1}{g}_{i}$ for 1≤i≤m$1le ile m$.如果在 G$G$ 中存在一个序列 g${/bf{g}}$，使得 G$G$ 的每个元素在 g${/bf{g}}$ 和 g¯$bar{/bf{g}}$ 中都恰好出现两次，那么我们就说 G$G$ 是双重可序列的。我们证明，如果一个群是无性的、奇数的、可序列的、R-可序列的或梯级可序列的，那么它就是双重可序列的。我们还证明，如果 H$H$ 是奇群或可排序群，而 K$K$ 是无边群，那么 H×K$Htimes K$ 是双重可排序的。

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引用次数: 0

Group divisible designs with block size 4 and group sizes 4 and 7 组块大小为 4，组块大小为 4 和 7 的可分割设计

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2024-03-28 DOI: 10.1002/jcd.21932

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

� � � 4 � � �

and group sizes

� � � 4 � � �

and

� � � 7 � � �

. We show that there exists a

� � � 4 � � �

-GDD of type

� � � �^{4 � t �} � �^{7 � s �} � � �

for all but a finite specified set of feasible values for

� � � � (� � t �, � s � �) � � � �

.

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 。

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引用次数: 0

Alternating parity weak sequencing 交替奇偶校验弱排序

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2024-02-27 DOI: 10.1002/jcd.21936

Simone Costa, Stefano Della Fiore

A subset <math> <semantics> <mrow> <mi>S</mi> </mrow> <annotation> $S$</annotation> </semantics></math> of a group <math> <semantics> <mrow> <mo>(</mo> <mrow> <mi>G</mi> <mo>,</mo> <mo>+</mo> </mrow> <mo>)</mo> </mrow> <annotation> $(G,+)$</annotation> </semantics></math> is <math> <semantics> <mrow> <mi>t</mi> </mrow> <annotation> $t$</annotation> </semantics></math>-weakly sequenceable if there is an ordering <math> <semantics> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>…</mi> <mo>,</mo> <msub> <mi>y</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo> </mrow> <annotation> $({y}_{1},{rm{ldots }},{y}_{k})$</annotation> </semantics></math> of its elements such that the partial sums <math> <semantics> <mrow> <msub> <mi>s</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>…</mi> <mo>,</mo> <msub> <mi>s</mi> <mi>k</mi> </msub> </mrow> <annotation> ${s}_{0},{s}_{1},{rm{ldots }},{s}_{k}$</annotation> </semantics></math>, given by <math> <semantics> <mrow> <msub> <mi>s</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> <annotation> ${s}_{0}=0$</annotation> </

群 (G,+)$(G,+)$ 的子集 S$S$ 是 t$t$ 弱可排序的，条件是其元素的排序 (y1,...,yk)$({y}_{1},{rm{ldots }},{y}_{k})$ 使得部分和 s0,s1,...,sk${s}_{0},{s}_{1},{rm{ldots }},{s}_{k}$, 给定 s0=0${s}_{0}=0$ 和 si=∑j=1iyj${s}_{i}={sum }_{j=1}^{i}{y}_{j}$ for 1≤i≤k$1le ile k$、满足 si≠sj${s}_{i}ne {s}_{j}$ 时，且 1≤∣i-j∣≤t$1le | i-j| le t$。Costa 等人证明，如果一个群的阶为 pe$pe$，那么当 p>3$pgt 3$ 是素数，e≤3$ele 3$ 和 t≤6$tle 6$ 时，所有足够大的非相同元素子集都是 t$t$ 弱可排序的。受这一结果的启发，我们证明，如果 G$G$ 是 Zp${{mathbb{Z}}}_{p}$ 和 Z2${{mathbb{Z}}}_{2}$ 的半间接积，且子集 S$S$ 是平衡的，那么 S$S$ 无论大小，只要 p>3$pgt 3$ 是素数，且 t≤8$tle 8$，就允许交替奇偶性 t$t$ 弱序列。如果 G$G$ 的一个子集包含相同数量的偶数元素和奇数元素，并且交替奇偶排序交替使用偶数元素和奇数元素，那么这个子集就是平衡的。然后，我们使用一种结合了拉姆齐理论和概率方法的混合方法，也证明了对于一般（非必要是非良性）群 N$N$ 和 Z2${{mathbb{Z}}}_{2}$ 的半直接乘积的群 G$G$，所有足够大的非相同元素平衡子集都接受交替奇偶性 t$t$ 弱排序。同样的程序也适用于研究一般的足够大（不一定平衡）集合的弱可排序性。在这里，我们已经能够证明，如果一个群 G$G$ 的子集 S$S$ 的大小足够大，并且如果 S$S$ 不包含 0，那么 S$S$ 是 t$t$ 弱可排序的。

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引用次数: 0

The existence of 2 3 ${2}^{3}$ -decomposable super-simple ( v , 4 , 6 ) $(v,4,6)$ -BIBDs 存在23个可分解的超简单（v,4,6）$(v,4,6)$-BIBDs

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2024-02-22 DOI: 10.1002/jcd.21935

Huangsheng Yu, Jingyuan Chen, R. Julian R. Abel, Dianhua Wu

A design is said to be super-simple if the intersection of any two blocks has at most two elements. A design with index <math> <semantics> <mrow> <mi>t</mi> <mi>λ</mi> </mrow> <annotation> $tlambda $</annotation> </semantics></math> is said to be <math> <semantics> <mrow> <msup> <mi>λ</mi> <mi>t</mi> </msup> </mrow> <annotation> ${lambda }^{t}$</annotation> </semantics></math>-decomposable, if its blocks can be partitioned into nonempty collections <math> <semantics> <mrow> <msub> <mi>B</mi> <mi>i</mi> </msub> </mrow> <annotation> ${{rm{ {mathcal B} }}}_{i}$</annotation> </semantics></math>, <math> <semantics> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>t</mi> </mrow> <annotation> $1le ile t$</annotation> </semantics></math>, such that each <math> <semantics> <mrow> <msub> <mi>B</mi> <mi>i</mi> </msub> </mrow> <annotation> ${{rm{ {mathcal B} }}}_{i}$</annotation> </semantics></math> with the point set forms a design with index <math> <semantics> <mrow> <mi>λ</mi> </mrow> <annotation> $lambda $</annotation> </semantics></math>. In this paper, it is proved that there exists a <math> <semantics> <mrow> <msup> <mn>2</mn> <mn>3</mn> </msup> </mrow> <annotation> ${2}^{3}$</annotation> </semantics></math>-decomposable super-simple <math> <semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $(v,4,6)$</annotation> </semantics></math>-BIBD (balanced incomplete block design) if and only if <math> <semantics> <mrow> <mi>v</mi> <mo>≥</mo>

如果任意两个图块的交集最多只有两个元素，则称该设计为超简单设计。如果一个有索引的设计的图块可以被分割为非空集合 , , , 这样每个集合与点集构成一个有索引的设计，那么这个设计就被称为可分解设计。本文证明，当且仅当和时，存在一个可分解的超简单 -BIBD（平衡不完全块设计）。

{"title":"The existence of \u0000 \u0000 \u0000 \u0000 2\u0000 3\u0000 \u0000 \u0000 ${2}^{3}$\u0000 -decomposable super-simple \u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 v\u0000 ,\u0000 4\u0000 ,\u0000 6\u0000 \u0000 )\u0000 \u0000 \u0000 $(v,4,6)$\u0000 -BIBDs","authors":"Huangsheng Yu, Jingyuan Chen, R. Julian R. Abel, Dianhua Wu","doi":"10.1002/jcd.21935","DOIUrl":"10.1002/jcd.21935","url":null,"abstract":"A design is said to be super-simple if the intersection of any two blocks has at most two elements. A design with index <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation> $tlambda $</annotation>\u0000 </semantics></math> is said to be <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>λ</mi>\u0000 <mi>t</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation> ${lambda }^{t}$</annotation>\u0000 </semantics></math>-decomposable, if its blocks can be partitioned into nonempty collections <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>B</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{rm{ {mathcal B} }}}_{i}$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>≤</mo>\u0000 <mi>i</mi>\u0000 <mo>≤</mo>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $1le ile t$</annotation>\u0000 </semantics></math>, such that each <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>B</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{rm{ {mathcal B} }}}_{i}$</annotation>\u0000 </semantics></math> with the point set forms a design with index <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation> $lambda $</annotation>\u0000 </semantics></math>. In this paper, it is proved that there exists a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mn>3</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation> ${2}^{3}$</annotation>\u0000 </semantics></math>-decomposable super-simple <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>,</mo>\u0000 <mn>4</mn>\u0000 <mo>,</mo>\u0000 <mn>6</mn>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(v,4,6)$</annotation>\u0000 </semantics></math>-BIBD (balanced incomplete block design) if and only if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>≥</mo>\u0000","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 6","pages":"297-307"},"PeriodicalIF":0.7,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139946643","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

Partition of ordered triples into uniform holey ordered designs 将有序三元组划分为均匀有序的洞穴设计

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2024-02-19 DOI: 10.1002/jcd.21933

Yuli Tan, Junling Zhou

A large set <math> <semantics> <mrow> <mtext>LOD</mtext> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> <annotation> $text{LOD}(v)$</annotation> </semantics></math> is a partition of all ordered triples of a <math> <semantics> <mrow> <mi>v</mi> </mrow> <annotation> $v$</annotation> </semantics></math>-set into <math> <semantics> <mrow> <mi>v</mi> <mo>−</mo> <mn>2</mn> </mrow> <annotation> $v-2$</annotation> </semantics></math> disjoint ordered designs of order <math> <semantics> <mrow> <mi>v</mi> </mrow> <annotation> $v$</annotation> </semantics></math>. In this paper, we generalize the large set <math> <semantics> <mrow> <mtext>LOD</mtext> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> <annotation> $text{LOD}(v)$</annotation> </semantics></math> with <math> <semantics> <mrow> <mi>v</mi> <mo>=</mo> <mi>g</mi> <mi>t</mi> </mrow> <annotation> $v=gt$</annotation> </semantics></math> to the notion of <math> <semantics> <mrow> <mtext>POT</mtext> <mrow> <mo>(</mo> <msup> <mi>g</mi> <mi>t</mi> </msup> <mo>)</mo> </mrow> </mrow> <annotation> $text{POT}({g}^{t})$</annotation> </semantics></math>, representing a partition of all ordered triples of a <math> <semantics> <mrow> <mi>g</mi> <mi>t</mi> </mrow> <annotation> $gt$</annotation> </semantics></math>-set into disjoint uniform holely ordered designs <math> <semantics> <mrow> <mtext>HOD</mtext> <mrow> <mo>(</mo> <msup> <mi>g</mi> <mi>t</mi> </msup> <mo>)</mo> </mrow> </mrow> <annotation> $text{HOD}({g}^{t})$</annotation> </semantics></math>s. We show that a <math> <semantics

大集合是将一个集合的所有有序三元组划分为秩为的互不相邻的有序设计。在本文中，我们将大集合的概念概括为，表示把一个集合的所有有序三元组分割成互不相交的均匀有序设计 s。此外，我们还研究了每个成员都有一种解析的 a 的存在性。我们证明，当且仅当、、、时，可解析的 a 存在，但有 27 个可能的例外。对于几乎可解的 s，我们证明了其渐近存在性，并提出了一些无穷族。

{"title":"Partition of ordered triples into uniform holey ordered designs","authors":"Yuli Tan, Junling Zhou","doi":"10.1002/jcd.21933","DOIUrl":"10.1002/jcd.21933","url":null,"abstract":"A large set <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>LOD</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>v</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{LOD}(v)$</annotation>\u0000 </semantics></math> is a partition of all ordered triples of a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math>-set into <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>−</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $v-2$</annotation>\u0000 </semantics></math> disjoint ordered designs of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math>. In this paper, we generalize the large set <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>LOD</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>v</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{LOD}(v)$</annotation>\u0000 </semantics></math> with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>=</mo>\u0000 <mi>g</mi>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $v=gt$</annotation>\u0000 </semantics></math> to the notion of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>POT</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>g</mi>\u0000 <mi>t</mi>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{POT}({g}^{t})$</annotation>\u0000 </semantics></math>, representing a partition of all ordered triples of a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>g</mi>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $gt$</annotation>\u0000 </semantics></math>-set into disjoint uniform holely ordered designs <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>HOD</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>g</mi>\u0000 <mi>t</mi>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{HOD}({g}^{t})$</annotation>\u0000 </semantics></math>s. We show that a <math>\u0000 <semantics","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 5","pages":"274-293"},"PeriodicalIF":0.7,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139946640","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