Blow-up lemmas for sparse graphs
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2016
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| _version_ | 1866915466657660928 |
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| author | Allen, Peter Böttcher, Julia Hàn, Hiep Kohayakawa, Yoshiharu Person, Yury |
| author_facet | Allen, Peter Böttcher, Julia Hàn, Hiep Kohayakawa, Yoshiharu Person, Yury |
| contents | 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 $Δ$ in subgraphs of $G(n,p)$ with $p=C(\log n/n)^{1/Δ}$; one for embedding spanning graphs with maximum degree $Δ$ and degeneracy $D$ in subgraphs of $G(n,p)$ with $p=C_Δ\big(\log n/n\big)^{1/(2D+1)}$; and one for embedding spanning graphs with maximum degree $Δ$ in $(p,cp^{\max(4,(3Δ+1)/2)}n)$-bijumbled graphs.
We also consider various applications of these lemmas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1612_00622 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Blow-up lemmas for sparse graphs Allen, Peter Böttcher, Julia Hàn, Hiep Kohayakawa, Yoshiharu Person, Yury Combinatorics 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 $Δ$ in subgraphs of $G(n,p)$ with $p=C(\log n/n)^{1/Δ}$; one for embedding spanning graphs with maximum degree $Δ$ and degeneracy $D$ in subgraphs of $G(n,p)$ with $p=C_Δ\big(\log n/n\big)^{1/(2D+1)}$; and one for embedding spanning graphs with maximum degree $Δ$ in $(p,cp^{\max(4,(3Δ+1)/2)}n)$-bijumbled graphs. We also consider various applications of these lemmas. |
| title | Blow-up lemmas for sparse graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/1612.00622 |