四元数$1-$分解与彩虹生成树的完备集
Quaternionic $1-$factorizations and complete sets of rainbow spanning trees
摘要：
一个完全图在$2n$顶点上的$1-$因子分解称为$G-$正则的，如果它有一个自同构群$G$在顶点集上严格传递.确定哪些群可以实现这样的情形的问题可以追溯到Hartman和Rosa（1985）关于循环群的结果，当$n$是偶数时，这个问题仍然没有解决。试图获得一个相当精确的描述群和$1-$因子分解满足这一对称性约束可以通过施加进一步的条件。最近证明了，参见Rinaldi（2021）和Mazzuoccolo等人（2019），只要$n$是奇数，就存在$G-$正则$1-$因子分解以及彩虹生成树的完整集合，而当$G$是循环的并且$n$不是$2$的幂时，或者当$G$是二面体群时，证明了每个$n$偶数的存在性。在本文中，我们推广了这一结果，并证明了存在性也适用于其他类别的群体。
英文摘要：
A $1-$factorization of a complete graph on $2n$ vertices is said to be $G-$regular if it posseses an automorphism group $G$ acting sharply transitively on the vertex-set. The problem of determining which groups can realize such a situation dates back to a result by Hartman and Rosa (1985) on cyclic groups and, when $n$ is even, the problem is still open. An attempt to obtain a fairly precise description of groups and $1-$factorizations satisfying this symmetry constrain can be done by imposing further conditions. It was recently proved, see Rinaldi (2021) and Mazzuoccolo et al. (2019), that a $G-$regular $1-$factorization together with a complete set of rainbow spanning trees exists whenever $n$ is odd, while the existence for each $n$ even was proved when either $G$ is cyclic and $n$ is not a power of $2$, or when $G$ is a dihedral group. In this paper we extend this result and prove the existence also for other classes of groups.
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。
