论文标题
Quaternionic $ 1- $ fractization和完整的彩虹跨越树木
Quaternionic $1-$factorizations and complete sets of rainbow spanning trees
论文作者
论文摘要
如果$ 2N $顶点的完整图表的$ 1- $分解，则如果它具有在顶点集合上急剧延伸的自动形态组$ g $，则为$ g- $常规。确定哪些群体可以意识到这种情况的问题可以追溯到Hartman和Rosa（1985）对循环群体的结果，而当$ n $的情况下，问题仍然开放。可以通过施加进一步的条件来实现对群体的相当精确描述和满足这种对称约束的$ 1-因素的尝试。最近证明了这一点，请参见Rinaldi（2021）和Mazzuoccolo等。 （2019年），每当$ n $奇怪时，就存在$ g-g $常规$ 1- $分解以及一套完整的彩虹跨越树木，而当$ g $ cyclic as Cyclic and $ n $不是$ 2 $的电源时，或者当$ n $的存在证明是$ 2 $时，或者当$ g $是$ g $时。在本文中，我们扩展了此结果，并证明了其他类别类别的存在。
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.
