$2$-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 $G$. In this article, among other results, we prove that $(k^2+2k+3)/4 \leqslant m(\mathbbmsl{O}_k,2)\leqslant (k^2+5k+3)/3$, where $\mathbbmsl{O}_k$ is the odd graph on a ground set of size $2k+1$. This confirms a conjecture posed in 2021 by Grippo, Pastine, Torres, Valencia-Pabon, and Vera.