$2$-Neighbor Bootstrap Percolation on Odd Graphs

Abstract

The $r$-neighbor bootstrap percolation process on a graph $G$ is a vertex-activation process that begins with a set of initially active vertices. In each subsequent round, every inactive vertex having at least $r$ active neighbors becomes active. Denote by $m(G,r)$ the minimum number of initially active vertices whose activation eventually spreads to all vertices of $G$. Let \(\mathbbmsl{O}_k\) and \(\mathbbmsl{B}_k\) be the odd graph and the bipartite odd graph, respectively, both defined on a ground set of size \(2k+1\). In this article, among other results, we prove that $m(\mathbbmsl{O}_k, 2)=\lceil(k^2+3k+2)/4\rceil$ and $m(\mathbbmsl{B}_k, 2)=\lceil(k^2+2k+3)/2\rceil$. The former result confirms a conjecture posed in 2021 by Grippo, Pastine, Torres, Valencia-Pabon, and Vera.