Partition of a graph into cycles and vertices

Let G be a graph of order n and k   a positive integer. A set of subgraphs H={H1,H2,…,Hk}H={H1,H2,…,Hk} is called a k-weak cycle partition (abbreviated k-WCP) of G   if H1,…,HkH1,…,Hk are vertex disjoint subgraphs of G   such that V(G)=⋃i=1kV(Hi) and for all i  , 1⩽i⩽k1⩽i⩽k, HiHi is a cycle or K1K1 or K2K2. It has been shown by Enomoto and Li that if |G|=n⩾k|G|=n⩾k and if the degree sum of any pair of nonadjacent vertices is at least n-k+1n-k+1, then G has a k-WCP. We prove that if G has a k  -WCP and if the minimum degree is at least (n+2k)/3(n+2k)/3, then G can be partitioned into k   subgraphs HiHi, 1⩽i⩽k1⩽i⩽k, where each HiHi is a cycle or K1K1.