Linear algebra and bootstrap percolation

Authors:
József Balogh;Béla Bollobás;Robert Morris;Oliver Riordan
Affiliations:
Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, United States and Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, Uni ...;Trinity College, Cambridge CB2 1TQ, England, United Kingdom and Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, United States;IMPA, Estrada Dona Castorina 110, Jardim Botínico, Rio de Janeiro, RJ, Brazil;Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, United Kingdom
Venue:
Journal of Combinatorial Theory Series A
Year:
2012

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Abstract

In H-bootstrap percolation, a set A@?V(H) of initially 'infected' vertices spreads by infecting vertices which are the only uninfected vertex in an edge of the hypergraph H. A particular case of this is the H-bootstrap process, in which H encodes copies of H in a graph G. We find the minimum size of a set A that leads to complete infection when G and H are powers of complete graphs and H encodes induced copies of H in G. The proof uses linear algebra, a technique that is new in bootstrap percolation, although standard in the study of weakly saturated graphs, which are equivalent to (edge) H-bootstrap percolation on a complete graph.