TITLE: Is every graph an isomorphic factor of some complete graph?
QUESTION [10 upvotes]: 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?

REPLY [11 votes]: 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 by J. Bosak might be helpful (the preview on google books is quite limited).