Given a sequence of nonnegative real numbers A,, A,, . . . which sum to 1, we consider random graphs having approximately Ain vertices of degree i. Essentially, we show that if C i(i -2)A, > 0, then such graphs almost surely have a giant component, while if C i(i -2)A, < 0, then almost surely all components in such graphs are small. We can apply these results to G n , p , Gn,M, and other well-known models of random graphs. There are also applications related to the chromatic number of sparse random graphs.

Given a sequence of non-negative r e a l n umbers 0 1 : : :which s u m t o 1 , w e consider a random graph having approximately i n vertices of degree i. In 12] the authors essentially show that if P i(i ; 2) i > 0 then the graph a.s. has a giant component, while if P i(i ; 2) i < 0 then a.s. all components in the graph are small. In this paper we analyze the size of the giant component in the former case, and the structure of the graph formed by deleting that component. We determine 0 0 0 1 : : : such that a.s. the giant component, C, h a s n + o ( n) v ertices, and the structure of the graph remaining after deleting C is basically that of a random graph with n 0 = n ; j Cj vertices, and with 0 i n 0 of them of degree i.

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