A Note on Weak Saturation Number of Trees

TL;DR

This paper investigates the weak saturation numbers of trees, especially caterpillars, providing estimates and characterizations for when trees are considered good based on their structure.

Contribution

It offers new estimates for weak saturation numbers of trees, characterizes good trees, and explores the relationship between tree structure and saturation properties.

Findings

Caterpillars have weak saturation numbers of order $k^eta$ for $eta ext{ in } [1,2]$.

A sufficient condition for a tree to be good is identified.

Trees with leaves at even distances are fully characterized as good trees under this condition.

Abstract

In this paper, we estimate the weak saturation numbers of trees. As a case study, we examine caterpillars and obtain several tight estimates. In particular, this implies that for any $\alpha\in [1,2]$, there exist caterpillars with $k$ vertices whose weak saturation numbers are of order $k^\alpha$. We call a tree good if its weak saturation number is exactly its edge number minus one. We provide a sufficient condition for a tree to be a good tree. With the additional property that all leaves are at even distances from each other, this condition fully characterizes good trees.