A decomposition of a graph $G$ into subgraphs $H$ is a collection of graphs all isomorphic to $H$ which are edge-disjoint in $G$ and together cover all the edges of $G$.
Let $u \geq 1$ and $v = u^2 - u + 1$.
I wish to show that there is a decomposition of $K_v - K_u$ into Hamilton paths.
As $v = u(u-1) + 1$ is odd, it is well-known that $K_v$ decomposes into Hamilton cycles.
Given a Hamilton cycle decomposition of $K_v$ it is sufficient to find a subset $U$ of $V$ containing $u$ vertices, such that one edge from every Hamilton cycle is contained in $U$. Is it always possible to find such a subset $U$? If not, may we find a Hamilton path decomposition of $K_v - K_u$ another way?