部分完美圈准支架的存在性

Some Results on the Perfect Cycle Semiframe

摘要: 设$K_{n,n,\cdots ,n}$ 是一个完全等$m$ 部图，若其边集全部能分解为$k$ 长的圈，且这些圈构成的集合既能划分成若干个平行类(即一组顶点互不相交的$k$ 长圈，且这组圈的顶点集的并恰好是整个点集的一个划分)，又能划分成若干个带洞平行类(即一组顶点互不相交的$k$ 长圈，且这组圈的顶点集的并恰好是除某一指定部(称为洞)外其余点集的一个划分)，则称该分解为一个长为$k$ 的完美圈准支架(perfect cycle semiframe, PCSF)，记为$\left(k,1\right)-PCSF\left(n^m\right)$ 。本文用循环群差分法寻找数值解，得到部分完美圈准支架的存在性。

Abstract: Let $K_{n,n,\cdots ,n}$ be a complete equipartite graph. If its edge set can be fully decomposed into cycles of length $k$ , and the collection of these cycles can be partitioned into several parallel classes (a set of vertex-disjoint cycles of length $k$ whose vertex union is exactly a partition of the entire vertex set), and can also be partitioned into several holey parallel classes (a set of vertex-disjoint cycles of length $k$ whose vertex union is exactly a partition of all vertices except for one specified part, called the hole), then such a decomposition is called a perfect cycle semiframe (PCSF) of length $k$ , denoted by $\left(k,1\right)-PCSF\left(n^m\right)$ . In this paper, we use the cyclic group difference method to seek numerical solutions, the existence of perfect cycle semiframes is obtained.