On $K_{2,t}$-bootstrap percolation

Abstract

Given two graphs $G$ and $H$, it is said that $G$ percolates in $H$-bootstrap process if one could join all the nonadjacent pairs of vertices of $G$ in some order such that a new copy of $H$ is created at each step. Balogh, Bollobás and Morris in 2012 investigated the threshold of $H$-bootstrap percolation in the Erdős-Rényi model for the complete graph $H$ and proposed the similar problem for $H=K_{s,t}$, the complete bipartite graph. In this paper, we provide lower and upper bounds on the threshold of $K_{2,t}$-bootstrap percolation. In addition, a threshold function is derived for $K_{2,4}$-bootstrap percolation.