Maximum Percolation Time on the q-ary Hypercube
Fengxing Zhu
TL;DR
This work determines the exact maximum percolation time for the 2-neighbor bootstrap percolation on the $n$-dimensional $q$-ary hypercube $Q_{n,q}$, extending Michal's binary-hypercube result to the nonbinary setting. Building on the Balogh–Bollobás framework, it introduces subcubes, closed sets, and nested sequences of internally spanned subcubes, coupled with six technical lemmas and norm-like measures to bound the infection spread. The authors establish a base casing $M_q(0)=0$, $M_q(1)=1$, $M_q(2)=3$, and derive explicit closed-form expressions for all $n\ge 3$: for $q=3$, $M_3(n)=\frac{n^2}{3}+\frac{2n}{3}$ with a $+\frac{1}{3}$ adjustment when $n\equiv 2\pmod{3}$; for $q\ge 4$, $M_q(n)=\frac{n^2}{3}+n$ with a $-\frac{1}{3}$ adjustment when $n\equiv 1,2\pmod{3}$. This completes the generalization of the maximum percolation-time results to $q$-ary hypercubes and frames several open questions, including higher thresholds and grid settings.
Abstract
We consider the $2$-neighbor bootstrap percolation process on the $n$-dimensional $q$-ary hypercube with vertex set $V=\{0,1,\dots,q-1\}^n$ and edges connecting the pairs at Hamming distance $1$. We extend the main theorem of Przykucki(2012) about the maximum percolation time with threshold $r=2$ on the binary hypercube to the $q$-ary case, finding the exact value of this time for all $q \geq 3$.