We study when a given edge of a factor-critical graph is contained in a matching avoiding exactly one, pregiven vertex of the graph. We then apply the results to always partition the vertex-set of a 3-regular, 3-uniform hypergraph into at most one triangle (hyperedge of size 3) and edges (subsets of size 2 of hyperedges), corresponding to the intuition, and providing new insight to triangle and edge packings of Cornuéjols’ and Pulleyblank’s. The existence of such a packing can be considered to be a hypergraph variant of Petersen’s theorem on perfect matchings, and leads to a simple proof for a sharpening of Lu’s theorem on antifactors of graphs.

我们研究了在匹配中包含关键因子图的给定边时，它恰好避免了图的一个预先指定的顶点的情况。然后，我们将结果应用于始终将3个正则3均匀超图的顶点集划分为与直觉相对应的最多一个三角形（尺寸为3的超边）和边缘（尺寸为2的超边的子集），并且为Cornuéjols和Pulleyblank的三角形和边缘填充提供了新的见解。这样的堆积的存在可以被认为是彼得森定理在完美匹配上的一个超图变体，并为简化图的反因子上的卢定理提供了简单的证明。