TY - JOUR

T1 - Long running times for hypergraph bootstrap percolation

AU - Espuny Díaz, Alberto

AU - Janzer, Barnabás

AU - Kronenberg, Gal

AU - Lada, Joanna

N1 - Publisher Copyright:
© 2023 The Author(s)

PY - 2024/1

Y1 - 2024/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85171674519&partnerID=8YFLogxK

U2 - 10.1016/j.ejc.2023.103783

DO - 10.1016/j.ejc.2023.103783

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AN - SCOPUS:85171674519

SN - 0195-6698

VL - 115

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

M1 - 103783

ER -