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 $K_4$? 

Here's the $(13,4,1)$-BIBD from the designtheory.org database (here), where it is described as unique up to isomorphism.

To check it's indeed a $(13,4,1)$-BIBD, we can run the following code in GAP:
S:=
[ [   1,   2,   3,   4 ],
  [   1,   5,   6,   7 ],
  [   1,   8,   9,  10 ],
  [   1,  11,  12,  13 ],
  [   2,   5,   8,  11 ],
  [   2,   6,   9,  12 ],
  [   2,   7,  10,  13 ],
  [   3,   5,   9,  13 ],
  [   3,   6,  10,  11 ],
  [   3,   7,   8,  12 ],
  [   4,   5,  10,  12 ],
  [   4,   6,   8,  13 ],
  [   4,   7,   9,  11 ] ];

A:=[];
for P in S do
  T:=Combinations(P,2);
  A:=Concatenation(A,T);
od;
Size(A)=Binomial(13,2);

which returns true.
