论文标题
相互正交的周期系统
Mutually orthogonal cycle systems
论文作者
论文摘要
图$γ$的$ {\ ell} $ - 循环系统$ {\ MATHCAL F} $是一组$ {\ ell} $ - 循环，该循环将$γ$的边缘集划分为$γ$。两个这样的周期系统$ {\ MATHCAL F} $和$ {\ MATHCAL F}'$被称为{\ em Orthogonal}，如果没有两个不同的循环与$ {\ Mathcal f} \ Cup {\ Mathcal F}'$分享比一边相比更多的循环。正交循环系统自然来自面部$ 2 $ - 颜色的Polyehdra，而在Heffter Arrays的较高属中，带有某些订单。 $γ$的一组成对的正交$ \ ell $ - 周期系统是$γ$的一组相互正交的周期系统。 令$μ（\ ell，n）$（分别为$μ'（\ ell，n）$）为最大整数$μ$，因此存在一组$μ$相互正交（循环）$ \ ell $ $ $ -CYCELE SYSTEMS完整图形$ k_n $。我们表明，如果$ \ ell \ geq 4 $均匀而$ n \ equiv 1 \ pmod {2 \ ell} $，则$μ'（\ ell，n）$，因此，$μ（\ ell，n）$在下面由$ n/\ ell^2 $的常数倍数。相比之下，我们获得以下上限：$μ（\ ell，n）\ leq n-2 $; $μ（\ ell，n）\ leq（n-2）（n-3）/（2（\ ell-3））$当$ \ ell \ geq 4 $; $μ（\ ell，n）\ leq 1 $当$ \ ell> n/\ sqrt {2} $;和$μ'（\ ell，n）\ leq n-3 $当$ n \ geq 4 $。我们还获得了$ n $和$ \ ell $的小值的计算结果。
An ${\ell}$-cycle system ${\mathcal F}$ of a graph $Γ$ is a set of ${\ell}$-cycles which partition the edge set of $Γ$. 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 $Γ$ is said to be a set of mutually orthogonal cycle systems of $Γ$. Let $μ(\ell,n)$ (respectively, $μ'(\ell,n)$) be the maximum integer $μ$ such that there exists a set of $μ$ 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 $μ'(\ell,n)$, and hence $μ(\ell,n)$, is bounded below by a constant multiple of $n/\ell^2$. In contrast, we obtain the following upper bounds: $μ(\ell,n)\leq n-2$; $μ(\ell,n)\leq (n-2)(n-3)/(2(\ell-3))$ when $\ell \geq 4$; $μ(\ell,n)\leq 1$ when $\ell>n/\sqrt{2}$; and $μ'(\ell,n)\leq n-3$ when $n \geq 4$. We also obtain computational results for small values of $n$ and $\ell$.
