Induced subgraphs in sparse random graphs with given degree sequences

Let Gn,dGn,d denote the uniformly random dd-regular graph on nn vertices. For any S⊂[n]S⊂[n], we obtain estimates of the probability that the subgraph of Gn,dGn,d induced by SS is a given graph HH. The estimate gives an asymptotic formula for any d=o(n1/3)d=o(n1/3), provided that HH 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.