Long running times for hypergraph bootstrap percolation

Espuny Díaz, Alberto, Janzer, Barnabás, Kronenberg, Gal and Lada, Joanna (2024) Long running times for hypergraph bootstrap percolation. European Journal of Combinatorics, 115. ISSN 0195-6698

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Abstract

Consider the hypergraph bootstrap percolation process in which, given a fixed r-uniform hypergraph H and starting with a given hypergraph G0, at each step we add to G0 all edges that create a new copy of H. We are interested in maximising the number of steps that this process takes before it stabilises. For the case where H=Kr+1(r) with r≥3, we provide a new construction for G0 that shows that the number of steps of this process can be of order Θ(nr). This answers a recent question of Noel and Ranganathan. To demonstrate that different running times can occur, we also prove that, if H is K4(3) minus an edge, then the maximum possible running time is 2n−⌊log2(n−2)⌋−6. However, if H is K5(3) minus an edge, then the process can run for Θ(n3) steps.

Item Type:	Article
Additional Information:	© 2023 The Author(s)
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	03 Oct 2023 23:27
Last Modified:	17 Nov 2024 00:21
URI:	http://eprints.lse.ac.uk/id/eprint/120360

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