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

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arxiv 2410.06132 v1 pith:FAE6R7Z6 submitted 2024-10-08 math.CO cs.DMmath.PR

classification math.COcs.DMmath.PR
keywords blow-uplemmagraphsspreadapplicationconjectureperturbedprove
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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 B\"ottcher, Parczyk, Sgueglia, and Skokan on the threshold for appearance of powers of Hamilton cycles in perturbed random graphs.

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  1. Robustness of the Sauer-Spencer Theorem

    math.CO 2025-07 accept novelty 8.0 of 10

    A random subgraph of a graph with minimum degree at least (1 - 1/(2Δ))n contains, with high probability, any spanning n-vertex graph of maximum degree Δ, once edges are kept with probability at least C n^{-1/m1(H)} log n.

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