Partition of a graph into cycles and vertices

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Let G be a graph of order n and k a positive integer. A set of subgraphs H={H1,H2,…,Hk} is called a k-weak cycle partition (abbreviated k-WCP) of G if H1,…,Hk are vertex disjoint subgraphs of G such that V(G)=⋃i=1kV(Hi) and for all i, 1⩽i⩽k, Hi is a cycle or K1 or K2. It has been shown by Enomoto and Li that if |G|=n⩾k and if the degree sum of any pair of nonadjacent vertices is at least n-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, then G can be partitioned into k subgraphs Hi, 1⩽i⩽k, where each Hi is a cycle or K1.