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arXiv · 2606.03687

$\boldsymbol{2}$-Neighbor Bootstrap Percolation on Odd Graphs

Ali Mohammadian·Sina Rezaie Zareie·Behruz Tayfeh-Rezaie
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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.

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Ali Mohammadian, Sina Rezaie Zareie, Behruz Tayfeh-Rezaie. 2026-06-02. $\boldsymbol{2}$-Neighbor Bootstrap Percolation on Odd Graphs. https://arxiv.org/abs/2606.03687

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