{"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 $\ell$.