Induced subgraphs in sparse random graphs with given degree sequences

Let G n , d denote the uniformly random d -regular graph on n vertices. For any S ⊂ [ n ] , we obtain estimates of the probability that the subgraph of G n , d induced by S is a given graph H . The estimate gives an asymptotic formula for any d = o ( n 1 / 3 ) , provided that H does not contain almost all the edges of the random graph. The result is further extended to the probability space of random graphs with a given degree sequence.