A complete graph is an undirected graph in which all pairs of vertices are connected by an edge. A perfect matching is a selection of edges of the graph which touch every vertex exactly once.
Is there a name for a set of perfect matchings of a complete graph that select every edge of the graph exactly once?
Background information for the interested:
One application of a perfect matching of a complete graph is sports scheduling: if vertices are teams, and edges are games between two teams, then the complete graph is a description of all unique pairings that is possible, and a perfect matching is a set of games that includes each team once. If you have $T$ teams, and $\frac{T}{2}$ fields, then the perfect matching describes games that can be played simultaneously in a single round.
If we want to ensure that each team eventually plays every other team, but no pair of teams play each other twice, we want to find a set of perfect matchings that select all edges of the graph exactly once.
For 8 teams, here are two different sets of perfect matchings that cover the space. The first row happened to be hand-crafted (hence the symmetry) while the second row is the result of round-robin scheduling:
Best Answer
It's called a 1-factorization. See for example, Wikipedia.