The existence of bipartite almost self-complementary 3-uniform hypergraphs

IF 1 Q1 MATHEMATICS
L. N. Kamble, C. Deshpande, B. Athawale
{"title":"The existence of bipartite almost self-complementary 3-uniform hypergraphs","authors":"L. N. Kamble, C. Deshpande, B. Athawale","doi":"10.7494/opmath.2023.43.5.663","DOIUrl":null,"url":null,"abstract":"An almost self-complementary 3-uniform hypergraph on \\(n\\) vertices exists if and only if \\(n\\) is congruent to 3 modulo 4 A hypergraph \\(H\\) with vertex set \\(V\\) and edge set \\(E\\) is called bipartite if \\(V\\) can be partitioned into two subsets \\(V_1\\) and \\(V_2\\) such that \\(e\\cap V_1\\neq \\emptyset\\) and \\(e\\cap V_2\\neq \\emptyset\\) for any \\(e\\in E\\). A bipartite self-complementary 3-uniform hypergraph \\(H\\) with partition \\((V_1, V_2)\\) of the vertex set \\(V\\) such that \\(|V_1|=m\\) and \\(|V_2|=n\\) exists if and only if either (i) \\(m=n\\) or (ii) \\(m\\neq n\\) and either \\(m\\) or \\(n\\) is congruent to 0 modulo 4 or (iii) \\(m\\neq n\\) and both \\(m\\) and \\(n\\) are congruent to 1 or 2 modulo 4. In this paper we define a bipartite almost self-complementary 3-uniform hypergraph \\(H\\) with partition \\((V_1, V_2)\\) of a vertex set \\(V\\) such that \\(|V_1|=m\\) and \\(|V_2|=n\\) and find the conditions on \\(m\\) and \\(n\\) for a bipartite 3-uniform hypergraph \\(H\\) to be almost self-complementary. We also prove the existence of bi-regular bipartite almost self-complementary 3-uniform hypergraphs.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Opuscula Mathematica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.7494/opmath.2023.43.5.663","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

An almost self-complementary 3-uniform hypergraph on \(n\) vertices exists if and only if \(n\) is congruent to 3 modulo 4 A hypergraph \(H\) with vertex set \(V\) and edge set \(E\) is called bipartite if \(V\) can be partitioned into two subsets \(V_1\) and \(V_2\) such that \(e\cap V_1\neq \emptyset\) and \(e\cap V_2\neq \emptyset\) for any \(e\in E\). A bipartite self-complementary 3-uniform hypergraph \(H\) with partition \((V_1, V_2)\) of the vertex set \(V\) such that \(|V_1|=m\) and \(|V_2|=n\) exists if and only if either (i) \(m=n\) or (ii) \(m\neq n\) and either \(m\) or \(n\) is congruent to 0 modulo 4 or (iii) \(m\neq n\) and both \(m\) and \(n\) are congruent to 1 or 2 modulo 4. In this paper we define a bipartite almost self-complementary 3-uniform hypergraph \(H\) with partition \((V_1, V_2)\) of a vertex set \(V\) such that \(|V_1|=m\) and \(|V_2|=n\) and find the conditions on \(m\) and \(n\) for a bipartite 3-uniform hypergraph \(H\) to be almost self-complementary. We also prove the existence of bi-regular bipartite almost self-complementary 3-uniform hypergraphs.
二部几乎自互补3-一致超图的存在性
上的几乎自互补3-一致超图 \(n\) 顶点存在当且仅当 \(n\) 等于3模4 A超图吗 \(H\) 有顶点集 \(V\) 边集 \(E\) 称为二部if \(V\) 可以分成两个子集吗 \(V_1\) 和 \(V_2\) 这样 \(e\cap V_1\neq \emptyset\) 和 \(e\cap V_2\neq \emptyset\) 对于任何 \(e\in E\). 二部自互补3-一致超图 \(H\) 有分区 \((V_1, V_2)\) 顶点集的 \(V\) 这样 \(|V_1|=m\) 和 \(|V_2|=n\) 存在当且仅当(i) \(m=n\) 或(ii) \(m\neq n\) 或者 \(m\) 或 \(n\) 等于0模4或(iii) \(m\neq n\) 两者都是 \(m\) 和 \(n\) 等于1或2模4。本文定义了一个二部几乎自互补3-一致超图 \(H\) 有分区 \((V_1, V_2)\) 顶点集的 \(V\) 这样 \(|V_1|=m\) 和 \(|V_2|=n\) 找到条件 \(m\) 和 \(n\) 对于二部3-一致超图 \(H\) 几乎是自我补充的。证明了双正则二部几乎自互补3-一致超图的存在性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Opuscula Mathematica
Opuscula Mathematica MATHEMATICS-
CiteScore
1.70
自引率
20.00%
发文量
30
审稿时长
22 weeks
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信