[Sharlin]

I was wondering the same thing. It does concern complete graphs only, but note also the additional requirement of edge-disjointedness (that is, every edge in the large graph is covered by exactly one copy of the subgraph).

>A typical decomposition question asks whether the edges of some graph G can be partitioned into disjoint copies of another graph H. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with n edges packs 2n+1 times into the complete graph K2n+1. In this paper, we prove this conjecture for large n.

https://arxiv.org/abs/2001.02665