Q1: 
For any given finite simple graph G with e edges, does there always exist an $n$ such that the edges of $K\_n$ can be partitioned into $\frac{\binom{n}{2}}{e}$ edge-disjoint copies of $G$? If so, can any upper bounds be placed on the minimal required $n$? 


Q2: 
Similarly for digraphs (with twice as many copies)?


Q3: 
For any loopless multidigraph with maximum edge multiplicity $m$, it seems clear that for some $M>m$, an $M$-complete multidigraph can be exactly partitioned into isomorphic copies of it, even without increasing $n$: if nothing smaller, one can always overlay $n!$ copies of the original multidigraph with each permutation of the vertices. Can this partition always be accomplished with $M=m$, increasing $n$ instead? If not, can tighter bounds be placed on $M$? 


Q4: Does allowing loops (and adjusting edge-counts appropriately) fundamentally change any of the above? 



Is every graph an isomorphic factor of some complete graph?
Q1: yes, this is a theorem by Wilson; see the first paragraph here: <https://arxiv.org/abs/1604.07282>


**Edit**: perhaps the book [Decomposition of graphs](https://books.google.com/books?hl=en&lr=&id=JQm7Nad3DuoC&oi=fnd&pg=PA3&ots=opWgR0u8Nd&sig=d1hkDAQ8kV7EVmyq6alqTLP43d4) by J. Bosak might be helpful (the preview on google books is quite limited).


