A Note on Weak Saturation Number of Trees
Wenchong Chen, Xiao-Chuan Liu, Xu Yang
TL;DR
This work studies weak saturation numbers for trees, exposing how w-sat(n,T) behaves independently of n for large n and can realize every rational exponent in [1,2] through tree constructions. It gives exact w-sat values for nondegenerate caterpillars, showing w-sat(T) = k-1 if the minimum pendant count a ≤ 2 and w-sat(T) = k-1 + {a-1 or2} otherwise, and identifies when such caterpillars are good trees. A central contribution is a structural characterization: among trees with even leaf distances, good trees are precisely those with a leaf adjacent to a degree-2 vertex, together with a counterexample to a prior theorem. The paper also develops general saturation bounds via end-star concepts, proves a key degree-based lemma, and demonstrates the existence of good trees with large second-smallest degrees, plus a growth-rate construction achieving w-sat = Theta(k^alpha) for any alpha in [1,2], highlighting both the breadth and the limitations of current saturation techniques.
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 $α\in [1,2]$, there exist caterpillars with $k$ vertices whose weak saturation numbers are of order $k^α$. 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.