Maximum running times for graph bootstrap percolation processes

• David Fabian ; Patrick Morris ^[1] ; Tibor Szabó ^[2]
  1. [1] Universitat Politècnica de Catalunya

     Universitat Politècnica de Catalunya

     Barcelona, España

     
  2. [2] Freie Universität Berlin, Germany
• Localización: Discrete Mathematics Days 2022 / Luis Felipe Tabera Alonso (ed. lit.), 2022, ISBN 978-84-19024-02-2, págs. 134-140
• Idioma: inglés
• Texto completo no disponible (Saber más ...)
• Resumen

  • Given a fixed graph H and an n-vertex graph G the H-bootstrap percolation process ofH on G is defined to be the sequence of graphs Gi, i ≥ 0 which starts with G0 := G andin which Gi+1 is obtained from Gi by adding every edge that completes a copy of H. Thisprocess is an example of a cellular automata and has been extensively studied since beingintroduced by Bollob´as in 1968. Recently, Bollob´as raised the question of determining themaximum running time of this process, over all choices of n-vertex graph G. Here, therunning time of the process is number of steps t the process takes before stabilising, thatis, when Gt = Gt+1. Recent papers of Bollob´as–Przykucki–Riordan–Sahasrabudhe, Matzkeand Balogh–Kronenberg–Pokrovskiy–Szab´o have addressed the case when H is a clique,and determined the asymptotics of this maximum running time for all cliques apart fromK5. Here, we initiate the study of the maximum running time for other graphs H andprovide a survey of our new results in this direction. We study several key examples, givingprecise results for trees and cycles, and giving general results towards understanding howthe maximum running time of the H-bootstrap percolation process depends on propertiesof H, in particular exploring the relationship between this graph parameter and the degreesequence of H. Many interesting questions remain and along the way, we indicate somedirections for future research.