Slow graph bootstrap percolation I: Cycles
David Fabian, Patrick Morris, Tibor Szabó
TL;DR
This work determines the exact maximum running time M_{C_k}(n) for the cycle H-bootstrap percolation process on n-vertex graphs, establishing M_{C_k}(n)=\\lceil \\log_{k-1}(n+k^2-4k+2)\\rceil if k is odd and M_{C_k}(n)=\\lceil \\log_{k-1}(2n+k^2-5k)\\rceil if k is even, for all sufficiently large n. The core insight is a diameter-diminishing mechanism: at each step the effective distance between vertices shrinks by a factor of k-1, yielding the leading logarithmic term, while the precise jump locations are controlled by Frobenius numbers associated with the semigroups generated by k-2 and k, namely F(k-2,k)=k^2-4k+2 and F'(k-2,k)=k^2/2-3k+2. The authors develop a path- and walk-based framework, including sumset analyses with sets D_i and A_i (and A'_i), to capture when edges appear, and combine this with parity arguments to resolve odd versus even k; these techniques enable exact running-time values and extend to disjoint unions of cycles with bounds depending on the largest cycle. This yields the first exact running-time results for an infinite family of nontrivial H and provides a structured approach for analyzing bootstrap percolation times in related graph families, with implications for cellular automata perspectives and extremal combinatorics.
Abstract
Given a fixed graph $H$ and an $n$-vertex graph $G$, the $H$-bootstrap percolation process on $G$ is defined to be the sequence of graphs $G_i$, $i\geq 0$ which starts with $G_0:=G$ and in which $G_{i+1}$ is obtained from $G_i$ by adding every edge that completes a copy of $H$. We are interested in $M_H(n)$ which is the maximum number of steps, over all $n$-vertex graphs $G$, that this process takes to stabilise. We determine this maximum running time precisely when $H$ is a cycle, giving the first infinite family of graphs $H$ for which an exact solution is known. We find that $M_{C_k}(n)$ is of order $\log_{k-1}(n)$ for all $3\leq k\in \mathbb{N}$. Interestingly though, the function exhibits different behaviour depending on the parity of $k$ and the exact location of the values of $n$ for which $M_H(n)$ increases is determined by the Frobenius number of a certain numerical semigroup depending on $k$.