Finding a random regular graph with degree d
The standard simple way of generating random regular graphs is:


while the degree &lt; d


choose a random perfect matching from the edges still possible to add to the graph
If no matching is possible, restart the process.



The problem with this is that the higher edge degree you want, the more likely it is for the algorithm to get stuck. I see many papers limiting themself to $|V|&gt;d^3$, so I don't know if this process will work for you.