Partitioning a Graph into Highly Connected Subgraphs

Given k ≥ 1, a k-proper partition of a graph G is a partition P of V (G) such that each part P of P induces a k-connected subgraph of G. We prove that if G is a graph of order n such that �(G) ≥ √ n, then G has a 2-proper partition with at most n/�(G) parts. The bounds on the number of parts and the minimum degree are both best possible. We then prove that If G is a graph of order n with minimum degree �(G) ≥ p c(k − 1)n, where c = 2123 180 , then G has a k-proper partition into at most cn δ(G) parts. This improves a result of Ferrara, Magnant and Wenger [Conditions for Families of Disjoint k-connected Subgraphs in a Graph, Discrete Math. 313 (2013), 760– 764] and both the degree condition and the number of parts are best possible up to the constant c.