A minimum-weight perfect matching pairs every vertex exactly once while minimizing total edge weight. General weighted perfect matching has polynomial-time algorithms. In the Christofides algorithm, it corrects precisely the odd-degree vertices of a minimum spanning tree. Alternating edges of a cyclic order on an even number of vertices give two perfect matchings; the cheaper one costs at most half that cycle's cost.
A one-factorization partitions every edge of a graph into perfect matchings. Each matching uses every vertex exactly once. A total of synthemes is precisely a one-factorization of the complete graph on six points.
