Degree conditions for the partition of a graph into cycles, edges and isolated vertices

Let k , n be integers with 2 ≤ k ≤ n , and let G be a graph of order n . We prove that if max { d G ( x ) , d G ( y ) } ≥ ( n − k + 1 ) / 2 for any x , y ∈ V ( G ) with x ≠ y and x y ∉ E ( G ) , then G has k vertex-disjoint subgraphs H 1 , … , H k such that V ( H 1 ) ∪ ⋯ ∪ V ( H k ) = V ( G ) and H i is a cycle or K 1 or K 2 for each 1 ≤ i ≤ k , unless k = 2 and G = C 5 , or k = 3 and G = K 1 ∪ C 5 .