Reference Request: "Resolutions" of $K_n$ for $n$ odd A resolution (in the combinatorial design sense) of $K_{n}$ is a collection of sets of edges of $K_{n}$ so that within each set of edges, each vertex appears once, and over the entire collection, each edge appears once. (If you prefer, it's a collection of complete matchings that together use each edge exactly once.)
For even $n$, there is a well-known construction of a resolution. Assuming the vertices are labelled $1,2, \ldots n$, define $X_{i}$ to be the set containing $(i,n)$, $(i-1,i+1)$, $(i-2,i+2)$, etc., reducing mod $n-1$ as necessary. This can be interpreted as a schedule for the games in a round-robin tournament - each set is the set of games to be played on a given day, and the properties of the resolution ensure each team will play once each day.
If $n$ is odd, a classical resolution is impossible. However, the tournament interpretation of the problem suggests a generalization - the problem now becomes to find a schedule of the games so that every team plays every other team exactly once, and every team plays exactly 2 games per day. We can adapt the algorithm above to solve this problem. We define $X_{i}$ to be the set containing $(i-1,i+1)$, $(i-2,i+2)$, $\ldots$, $(i -\frac{n-1}{2},i+ \frac{n-1}{2})$, reducing mod $n$ as needed. Then $X_{1}$ contains every vertex except 1. For each pair $(x,y)$ in $X_{1}$, we can form the set $Y_{(x,y)} = X_{x} \cup X_{y} \cup \{(x,y)\}$. It is easy to check that the sets $Y_{(x,y)}$ partition the edges of $K_{n}$, and each one contains each vertex exactly twice, solving the problem.
I would really like to find a reference for the odd case here. Every book on tournaments I've found proves the even case, but I haven't seen any discussion of the generalization to the odd case. (And I would be shocked if I were the first one to have noticed it.) Any help would be appreciated. 
 A: I can't find an explicit reference to this, however an odd balanced tournament design (see Exercise 3 on p206 of Ian Anderson's Combinatorial Designs and Tournaments, Oxford 1997) has similarities.  Briefly this starts from the classic construction of a resolvable tournament design for an even number of teams (see p13 of the same book), and then deletes all the games that involve the infinite element.  What's left is a design where teams play exactly twice at each venue. So taking this construction and swapping venues and days around, gives a schedule like yours where teams play exactly twice per day.
A: I might be missing something, but it looks like you want a partition of the edges of $K_n$, odd $n$, into a set of subgraphs which are regular of degree 2.  That's called a 2-factorization. It can be done in very many ways, for example take the sets $$X_i = \{ (j,j+i)\mathrel: 1\le j\le n\}$$ for $1\le i\le (n-1)/2$.  It is also possible to choose the subgraphs to be connected, in which case it is called a hamiltonian decomposition.
