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arXiv · 1612.00622

Blow-up lemmas for sparse graphs

Peter Allen·Julia Böttcher·Hiep Hàn·Yoshiharu Kohayakawa·Yury Person
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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.

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Peter Allen, Julia Böttcher, Hiep Hàn, Yoshiharu Kohayakawa, Yury Person. 2016-12-02. Blow-up lemmas for sparse graphs. https://doi.org/10.19086/da.143410

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