A critical point for random graphs with a given degree sequence

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