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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.08465 |
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Table of Contents:
- A vertex partition in which every part induces a 2-connected subgraph is called a 2-proper partition. This concept was introduced by Ferrara et al. in 2013, and Borozan et al. gave the best possible minimum degree condition for the existence of a 2-proper partition in 2016. Later, in 2022, Chen et al. extended the result by showing a minimum degree sum condition for the existence of 2-proper partition. In this paper, we introduce two new invariants of graph, denoted by $σ^*(G)$ and $α^*(G)$. These two invariants are defined from degree sum on all independent sets with some property. We prove that if a graph $G$ satisfies $σ^*(G)\geq |V(G)|$, then with some exceptions, $G$ has a 2-proper partition with at most $α^*(G)$ parts. This result is best possible, and implies both of the results by Borozan et al. and by Chen et al.. Moreover, as a corollary of our result, we give a minimum degree product condition for the existence of a 2-proper partition.