A Note on Weak Saturation Number of Trees

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. The latter result also provides counterexamples, demonstrating that Theorem 8 of a paper by Faudree, Gould and Jacobson (R. J. Faudree, R. J. Gould, and M. S. Jacobson. Weak saturation numbers for sparse graphs. {\it Discussiones Mathematicae Graph Theory}, 33(4): 677-693, 2013.) is incorrect.