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. 



Reference Request: "Resolutions" of $K_n$ for $n$ odd
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.


