Slow graph bootstrap percolation I: Cycles

TL;DR

This paper precisely determines the maximum number of steps for the slow graph bootstrap percolation process on graphs when the fixed graph is a cycle, revealing a logarithmic growth pattern influenced by the cycle length and parity.

Contribution

It provides the first exact solution for the maximum stabilization time in bootstrap percolation when the fixed graph is a cycle, including detailed behavior based on cycle length parity.

Findings

Maximum steps grow logarithmically with the number of vertices.

Behavior varies depending on the parity of the cycle length.

The increase points are linked to the Frobenius number of a numerical semigroup.

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 $k$.