On percolation in random graphs with given vertex degrees
  We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for random graphs with given vertex degrees. This is used to study existence of giant component and existence of k-core. As a variation of the latter, we study also bootstrap percolation in random regular graphs.   We obtain both simple new proofs of known results and new results. An interesting feature is that for some degree sequences, there are several or even infinitely many phase transitions for the k-core. 
Svante Janson