Minimum Conditions for Bootstrap Percolation on the Cubic Graph

Abstract

Bootstrap percolation is an iterative process on the vertices of a graph. Initially, a proper, non-empty set of vertices is infected, and all other vertices are uninfected. At each iteration, every uninfected vertex with a certain number of infected neighbors becomes infected, and all infected vertices remain so permanently. At the end of the process, if all vertices are infected, percolation occurs. In this case, the initial set of infected vertices percolates the graph. Necessary and sufficient conditions for the minimum size of a percolating set and the minimum number of rounds to achieve percolation on a cubic graph of order 2n are presented, for any integer n, 2n ≥ 4.