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Spread blow-up lemma with an application to perturbed random graphs

Rajko Nenadov, Huy Tuan Pham

TL;DR

The paper proves a spread version of the blow-up lemma by combining algorithmic embedding techniques with spread perfect matchings, yielding an $O(1/N)$-spread distribution over embeddings of a spanning graph $H$ into systems of $(\varepsilon,\delta)$-super-regular pairs. This spread blow-up lemma enables probabilistic threshold reasoning via the Kahn–Kalai framework and is applied to perturbed random graphs to establish an approximate threshold for the appearance of the $k$-th power of Hamilton cycles in $G\cup G(n,p)$ under a typical minimum-degree regime. The main technical contributions are the construction of a carefully phased embedding algorithm with quasirandom control and a vertex-spread analysis that leverages recent results on spread perfect matchings. The results bridge the regularity method with random perturbations and provide robust tools for embedding problems in dense-plus-random graph models, with potential to tighten threshold results further in the perturbed setting.

Abstract

Combining ideas of Pham, Sah, Sawhney, and Simkin on spread perfect matchings in super-regular bipartite graphs with an algorithmic blow-up lemma, we prove a spread version of the blow-up lemma. Intuitively, this means that there exists a probability measure over copies of a desired spanning graph $H$ in a given system of super-regular pairs which does not heavily pin down any subset of vertices. This allows one to complement the use of the blow-up lemma with the recently resolved Kahn-Kalai conjecture. As an application, we prove an approximate version of a conjecture of Böttcher, Parczyk, Sgueglia, and Skokan on the threshold for appearance of powers of Hamilton cycles in perturbed random graphs.

Spread blow-up lemma with an application to perturbed random graphs

TL;DR

The paper proves a spread version of the blow-up lemma by combining algorithmic embedding techniques with spread perfect matchings, yielding an -spread distribution over embeddings of a spanning graph into systems of -super-regular pairs. This spread blow-up lemma enables probabilistic threshold reasoning via the Kahn–Kalai framework and is applied to perturbed random graphs to establish an approximate threshold for the appearance of the -th power of Hamilton cycles in under a typical minimum-degree regime. The main technical contributions are the construction of a carefully phased embedding algorithm with quasirandom control and a vertex-spread analysis that leverages recent results on spread perfect matchings. The results bridge the regularity method with random perturbations and provide robust tools for embedding problems in dense-plus-random graph models, with potential to tighten threshold results further in the perturbed setting.

Abstract

Combining ideas of Pham, Sah, Sawhney, and Simkin on spread perfect matchings in super-regular bipartite graphs with an algorithmic blow-up lemma, we prove a spread version of the blow-up lemma. Intuitively, this means that there exists a probability measure over copies of a desired spanning graph in a given system of super-regular pairs which does not heavily pin down any subset of vertices. This allows one to complement the use of the blow-up lemma with the recently resolved Kahn-Kalai conjecture. As an application, we prove an approximate version of a conjecture of Böttcher, Parczyk, Sgueglia, and Skokan on the threshold for appearance of powers of Hamilton cycles in perturbed random graphs.
Paper Structure (19 sections, 10 theorems, 26 equations)

This paper contains 19 sections, 10 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Proof of the spread blow-up lemma
  3. Preliminaries
  4. Algorithm
  5. Pre-processing.
  6. Quasirandom embedding.
  7. Embedding algorithm.
  8. Phase I: Embed $H \setminus B$ (and maybe some vertices from $B$).
  9. Phase II: Embed $B$.
  10. Proof of correctness and vertex-spread
  11. Early stage: $j < |N_H(B)|$.
  12. Exceptional stage: $j = |N_H(B)|$.
  13. Regular stage: $j > |N_H(B)|$.
  14. Phase II.
  15. Vertex-spread.
  16. ...and 4 more sections

Key Result

Lemma 2

For every $r, \Delta \in \mathbb{N}$ and $\delta, \alpha > 0$ there exist $\varepsilon, \beta > 0$ such that the following holds: Then there exists an $O(1/N)$-vertex-spread distribution $\lambda$ over embeddings $\phi \colon H \hookrightarrow G$ with the property that $\phi(x) \in W_x$ for every $x \in V(H)$.

Theorems & Definitions (26)

  • Definition 1
  • Lemma 2: Spread Blow-up Lemma
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • proof
  • Theorem 6
  • Lemma 7
  • Definition 8
  • Claim 9
  • ...and 16 more