A Note on Weak Saturation Number of Trees
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911558941016064 |
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| author | Chen, Wenchong Liu, Xiao-Chuan Yang, Xu |
| author_facet | Chen, Wenchong Liu, Xiao-Chuan Yang, Xu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15626 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Note on Weak Saturation Number of Trees Chen, Wenchong Liu, Xiao-Chuan Yang, Xu Combinatorics 05C35 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. |
| title | A Note on Weak Saturation Number of Trees |
| topic | Combinatorics 05C35 |
| url | https://arxiv.org/abs/2502.15626 |