Blow-up lemmas for sparse graphs

TL;DR

This paper extends the classical blow-up lemma to sparse graphs, enabling the embedding of bounded degree spanning graphs into random and pseudorandom graphs with specific probabilistic and pseudorandom conditions.

Contribution

It introduces three new sparse versions of the blow-up lemma applicable to random and pseudorandom graphs, broadening their utility in extremal combinatorics.

Findings

Successful embedding of spanning graphs with bounded degree in sparse random graphs.

Extension of blow-up lemma to pseudorandom graphs with bijumbledness.

Applicability of the lemmas to various combinatorial problems.

Abstract

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 $\Delta$ in subgraphs of $G(n,p)$ with $p=C(\log n/n)^{1/\Delta}$; one for embedding spanning graphs with maximum degree $\Delta$ and degeneracy $D$ in subgraphs of $G(n,p)$ with $p=C_\Delta\big(\log n/n\big)^{1/(2D+1)}$; and one for embedding spanning graphs with maximum degree $\Delta$ in $(p,cp^{\max(4,(3\Delta+1)/2)}n)$-bijumbled graphs. We also consider various applications of these lemmas.