2-Proper Partition of a Graph

Abstract

For a graph G,  if $\mathcal {P}(G)=\{H_1,H_2,\ldots ,H_t\}$ is a set of pairwise disjoint subsets of V(G) such that $V(G)=\bigcup \nolimits _{i=1}^{t}V(H_i)$ and each subgraph induced by $H_i$ is k-connected, $1\le i\le t,$ then $\mathcal {P}(G)$ is called a k-proper partition of G. Let $\delta (G)=\min \{d(v):v\in V(G)\}$ and $\sigma (G)=\min \{d(x)+d(y):xy\notin E(G),x,y\in V(G)\}.$ Borozan et.al [J. Graph Theory 82(2016)] proved that if G is a graph of order n with $\delta (G)\ge \sqrt{n},$ then G has a 2-proper partition $\mathcal {P}(G)$ with $|\mathcal {P}(G)|\le \frac{n-1}{\delta (G)}.$ In this paper, we prove that for a graph G of order n,  if $\delta (G)\ge \sqrt{n},$ or $1\le \delta (G)\le \sqrt{n}-1$ and $\sigma (G)\ge \frac{n}{\delta (G)}+\delta (G)-1,$ then G has a 2-proper partition $\mathcal {P}(G)$ with $|\mathcal {P}(G)|\le \frac{2(n-1)}{\sigma (G)}.$ The bounds of $\delta (G)$ and $\sigma (G)$ in our result are best possible.