二次拉丁方的循环与反完美1-因子分解
Cycles of quadratic Latin squares and anti-perfect $1$-factorisations
AI总结：
本文研究二次拉丁方的$2\times2$子方特征，并用其构造完全图与完全二部图的反完美1-因子分解，同时限制可产生完美1-因子分解的素数幂阶。
摘要：
阶为$n$的拉丁方是由$n$个符号组成的$n\times n$矩阵，使得每个符号在每一行和每一列中恰好出现一次。对于奇素数幂$q$，令$\mathbb{F}_q$表示阶为$q$的有限域。二次拉丁方是定义为$\mathcal{L}[a,b]$的拉丁方：当$j-i$是$\mathbb{F}_q$中的二次剩余时，$(\mathcal{L}[a,b])_{i,j}=i+a(j-i)$；否则为$i+b(j-i)$，其中$\{a,b\}\subseteq\mathbb{F}_q$，且$ab$和$(a-1)(b-1)$都是$\mathbb{F}_q$中的二次剩余。二次拉丁方此前已被用于构造完美1-因子分解、相互正交拉丁方和原子拉丁方。我们首先刻画不含$2\times2$拉丁子方的二次拉丁方。设$G$为图，$\mathcal{F}$为$G$的1-因子分解。如果$\mathcal{F}$中每一对1-因子的并都在$G$中诱导出一条Hamilton圈，则称$\mathcal{F}$是完美的；如果$\mathcal{F}$中不存在任何一对1-因子在$G$中诱导出Hamilton圈，则称$\mathcal{F}$是反完美的。我们利用二次拉丁方构造完全图和完全二部图的反完美1-因子分解的新例子。我们还证明，对于每个奇素数$p$，只有有限多个作为$p$的幂的阶$q$，使得阶为$q$的二次拉丁方可用于构造完全图或完全二部图的完美1-因子分解。
英文摘要：
A Latin square of order $n$ is an $n \times n$ matrix of $n$ symbols, such that each symbol occurs exactly once in each row and column. For an odd prime power $q$ let $\mathbb{F}_q$ denote the finite field of order $q$. A quadratic Latin square is a Latin square $\mathcal{L}[a, b]$ defined by, $$(\mathcal{L}[a, b])_{i, j} = \begin{cases} i + a(j-i) & \text{if } j-i \text{ is a quadratic residue in } \mathbb{F}_q, \\ i + b(j-i) & \text{otherwise}, \end{cases}$$ for some $\{a, b\} \subseteq \mathbb{F}_q$ such that $ab$ and $(a-1)(b-1)$ are quadratic residues in $\mathbb{F}_q$. Quadratic Latin squares have previously been used to construct perfect $1$-factorisations, mutually orthogonal Latin squares and atomic Latin squares. We first characterise quadratic Latin squares which are devoid of $2 \times 2$ Latin subsquares. Let $G$ be a graph and $\mathcal{F}$ a $1$-factorisation of $G$. If the union of every pair of $1$-factors in $\mathcal{F}$ induces a Hamiltonian cycle in $G$ then $\mathcal{F}$ is called perfect, and if there is no pair of $1$-factors in $\mathcal{F}$ which induce a Hamiltonian cycle in $G$ then $\mathcal{F}$ is called anti-perfect. We use quadratic Latin squares to construct new examples of anti-perfect $1$-factorisations of complete graphs and complete bipartite graphs. We also demonstrate that for each odd prime $p$, there are only finitely many orders $q$, which are powers of $p$, such that quadratic Latin squares of order $q$ could be used to construct perfect $1$-factorisations of complete graphs or complete bipartite graphs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
