I have the following problem:
Given a complete, simple graph, i.e., a graph $(V,E)$ where every possible edge is realized, so $|E| = \frac{|V||V-1|}2$. Now consider complete sub-graphs, or sub-cliques, of size $k$ be chosen at random. Every times a sub-clique $(V',E')$ is chosen, the edges $E'\subseteq E$ are colored. How long does it take in expectation until every edge in $E$ is colored?
I have searched for this problem, but not found anything. Was this discussed before or does this problem even have a name? For the case of $k=2$ it can be seen as the coupon collector's problem with $|E|$ coupons, but for bigger $k$ I found nothing.
PS: I am not asking for a solution, only for a direction, if there is a published solution ;)