Blow-up lemmas for sparse graphs

The blow-up lemma states that a system of super-regular pairs contains all bounded degree spanning graphs as subgraphs that embed into a corresponding system of complete pairs. This lemma has far-reaching applications in extremal combinatorics. We prove sparse analogues of the blow-up lemma for subgraphs of random and of pseudorandom graphs. Our main results are the following three sparse versions of the blow-up lemma: one for embedding spanning graphs with maximum degree ∆ in subgraphs of G(n, p) with p = C(logn/n) 1/∆ ; one for embedding spanning graphs with maximum degree ∆ and degeneracy D in subgraphs of G(n, p) with p = C (logn/n) 1/(2D+1) ; and one for embedding spanning graphs with maximum degree ∆ in (p, cpmax(4,(3∆+1)/2)n)-bijumbled graphs. We also consider various applications of these lemmas