Graph decomposition What is the smallest $n \in \mathbb{N}$ with $ n \geq5$ such that the edge set of the complete graph $K_n$ can be partitioned (decomposed) to edge disjoint copies of $K_4$?
I got a necesary condition for the decomposition is that $12 |n(n-1)$ and $3|n-1$, thus it implies $n \geq 13$. But can $K_{13}$ indeed be decomposed into edge disjoint copies of $K_4$? 

The degree of $K_9$ is 8, whereas the degree of $K_4$ is 3. Since $3$ does not divide $8$, there is no $K_4$ decomposition of $K_9$.
$K_n$ has a decomposition into edge-disjoint copies of $K_4$ whenever $n \equiv 1 \text{ or 4 } (\text{mod} 12)$, so the next smallest example after $K_4$ is $K_{13}$.
