Optimal and Near-Optimal Constructions for Bootstrap Percolation in Hypercubes

2026-04-16

AI Summary

This study addresses the extremal problem of 4-neighbor bootstrap percolation on the $d$-dimensional hypercube graph $Q_d$, aiming to determine the minimum number $m(Q_d;4)$ of initially infected vertices required to eventually infect the entire graph. By combining combinatorial constructions, extremal graph theory arguments, and AlphaEvolve-assisted search, the authors establish for the first time that the Morrison–Noel lower bound is tight for infinitely many dimensions. They further provide a general upper bound that differs from this lower bound by only $O(d)$. A key contribution is the exact formula $m(Q_d;4) = \frac{d(d^2 + 3d + 14)}{24} + 1$, which holds for infinitely many values of $d$, substantially advancing the understanding of the $r=4$ bootstrap percolation regime on hypercubes.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The $r$-neighbour bootstrap process on a graph $G$ begins with a set of infected vertices; subsequently, healthy vertices become infected once they have at least $r$ infected neighbours. The central extremal problem in bootstrap percolation is to determine the minimum cardinality of an initial infected set that eventually spreads to all vertices of $G$, denoted $m(G;r)$. Morrison and Noel established a general lower bound on $m(Q_d;r)$, where $Q_d$ is the $d$-dimensional hypercube, and asked whether it is tight whenever $d$ is sufficiently large with respect to $r$. This question was answered affirmatively for $r\leq 3$. In this paper, we show that $m(Q_d;4)=\frac{d(d^2+3d+14)}{24}+1$, matching the bound in of Morrison and Noel, for infinitely many $d$. We also obtain, for general $d$, an upper bound on $m(Q_d;4)$ that differs from the Morrison--Noel lower bound by an additive $O(d)$ term. Several key constructions in this paper were obtained with the assistance of AlphaEvolve.

Problem

• bootstrap percolation
• hypercubes
• extremal problem
• initial infected set
• minimum cardinality

Innovation

• bootstrap percolation
• hypercubes
• extremal combinatorics
• AlphaEvolve
• infection threshold