New Invariants for Partitioning a Graph Into 2-Connected Subgraphs

Abstract

Let $G$ be a graph of order $n$. For an integer $k\ge 2$, a partition $P$ of $V\left(G\right)$ is called a $k$-proper partition of $G$ if every $P\in P$ induces a $k$-connected subgraph of $G$. This concept was introduced by Ferrara et al., and Borozan et al. gave minimum degree conditions for the existence of a $k$-proper partition. In particular, when $k=2$, they proved that if $\delta \left(G\right)\ge \sqrt{n}$, then $G$ has a 2-proper partition $P$ with $\mid P\mid \le \frac{n-1}{\delta \left(G\right)}$. Later, Chen et al. extended the result by giving a minimum degree sum condition for the existence of a 2-proper partition. In this paper, we introduce two new invariants of graphs ${\sigma }^{*}\left(G\right)$ and ${\alpha }^{*}\left(G\right)$, which are defined from degree sum of particular independent sets. Our result is that if ${\sigma }^{*}\left(G\right)\ge n$, then with some exceptions, $G$ has a 2-proper partition $P$ with $\mid P\mid \le {\alpha }^{*}\left(G\right)$. We completely determine exceptional graphs. This result implies both of results by Borozan et al. and by Chen et al. Moreover, we obtain a minimum degree product condition for the existence of a 2-proper partition as a corollary of our result.