Mutually orthogonal cycle systems

A. Burgess, Nicholas J. Cavenagh, David A. Pike
{"title":"Mutually orthogonal cycle systems","authors":"A. Burgess, Nicholas J. Cavenagh, David A. Pike","doi":"10.26493/1855-3974.2692.86d","DOIUrl":null,"url":null,"abstract":"An ${\\ell}$-cycle system ${\\mathcal F}$ of a graph $\\Gamma$ is a set of ${\\ell}$-cycles which partition the edge set of $\\Gamma$. Two such cycle systems ${\\mathcal F}$ and ${\\mathcal F}'$ are said to be {\\em orthogonal} if no two distinct cycles from ${\\mathcal F}\\cup {\\mathcal F}'$ share more than one edge. Orthogonal cycle systems naturally arise from face $2$-colourable polyehdra and in higher genus from Heffter arrays with certain orderings. A set of pairwise orthogonal $\\ell$-cycle systems of $\\Gamma$ is said to be a set of mutually orthogonal cycle systems of $\\Gamma$. Let $\\mu(\\ell,n)$ (respectively, $\\mu'(\\ell,n)$) be the maximum integer $\\mu$ such that there exists a set of $\\mu$ mutually orthogonal (cyclic) $\\ell$-cycle systems of the complete graph $K_n$. We show that if $\\ell\\geq 4$ is even and $n\\equiv 1\\pmod{2\\ell}$, then $\\mu'(\\ell,n)$, and hence $\\mu(\\ell,n)$, is bounded below by a constant multiple of $n/\\ell^2$. In contrast, we obtain the following upper bounds: $\\mu(\\ell,n)\\leq n-2$; $\\mu(\\ell,n)\\leq (n-2)(n-3)/(2(\\ell-3))$ when $\\ell \\geq 4$; $\\mu(\\ell,n)\\leq 1$ when $\\ell>n/\\sqrt{2}$; and $\\mu'(\\ell,n)\\leq n-3$ when $n \\geq 4$. We also obtain computational results for small values of $n$ and $\\ell$.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"219 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2022-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ars Math. Contemp.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.26493/1855-3974.2692.86d","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 5

Abstract

An ${\ell}$-cycle system ${\mathcal F}$ of a graph $\Gamma$ is a set of ${\ell}$-cycles which partition the edge set of $\Gamma$. Two such cycle systems ${\mathcal F}$ and ${\mathcal F}'$ are said to be {\em orthogonal} if no two distinct cycles from ${\mathcal F}\cup {\mathcal F}'$ share more than one edge. Orthogonal cycle systems naturally arise from face $2$-colourable polyehdra and in higher genus from Heffter arrays with certain orderings. A set of pairwise orthogonal $\ell$-cycle systems of $\Gamma$ is said to be a set of mutually orthogonal cycle systems of $\Gamma$. Let $\mu(\ell,n)$ (respectively, $\mu'(\ell,n)$) be the maximum integer $\mu$ such that there exists a set of $\mu$ mutually orthogonal (cyclic) $\ell$-cycle systems of the complete graph $K_n$. We show that if $\ell\geq 4$ is even and $n\equiv 1\pmod{2\ell}$, then $\mu'(\ell,n)$, and hence $\mu(\ell,n)$, is bounded below by a constant multiple of $n/\ell^2$. In contrast, we obtain the following upper bounds: $\mu(\ell,n)\leq n-2$; $\mu(\ell,n)\leq (n-2)(n-3)/(2(\ell-3))$ when $\ell \geq 4$; $\mu(\ell,n)\leq 1$ when $\ell>n/\sqrt{2}$; and $\mu'(\ell,n)\leq n-3$ when $n \geq 4$. We also obtain computational results for small values of $n$ and $\ell$.
互正交循环系
图$\Gamma$的${\ell}$ -循环系统${\mathcal F}$是一组${\ell}$ -循环,它们划分了$\Gamma$的边集。两个这样的循环系统${\mathcal F}$和${\mathcal F}'$是{\em正交}的,如果${\mathcal F}\cup {\mathcal F}'$上没有两个不同的循环共享一条以上的边。正交循环系统自然地从面$2$ -可着色的聚hhdra和具有一定顺序的Heffter数组的更高属中产生。一组$\Gamma$的对正交$\ell$ -循环系统称为$\Gamma$的互正交循环系统的集合。设$\mu(\ell,n)$(分别为$\mu'(\ell,n)$)为最大整数$\mu$,使得完全图$K_n$存在一组$\mu$相互正交(循环)$\ell$ -循环系统。我们证明,如果$\ell\geq 4$是偶数并且$n\equiv 1\pmod{2\ell}$,那么$\mu'(\ell,n)$,因此$\mu(\ell,n)$,以$n/\ell^2$的常数倍为界。相反,我们得到以下上界:$\mu(\ell,n)\leq n-2$;$\mu(\ell,n)\leq (n-2)(n-3)/(2(\ell-3))$当$\ell \geq 4$;$\mu(\ell,n)\leq 1$当$\ell>n/\sqrt{2}$;当$n \geq 4$的时候$\mu'(\ell,n)\leq n-3$。我们也得到了小值$n$和$\ell$的计算结果。
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
自引率
0.00%
发文量
0
×
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