Covering vertices by a specified number of disjoint cycles, edges and isolated vertices

Let n , k be integers with n ≥ k ≥ 2 , and let G be a graph of order n and S be a subset of V ( G ) . We define μ ( S ) : = min { max { d G ( x ) , d G ( y ) } ∣ x , y ∈ S , x ≠ y , x y ∉ E ( G ) } if there exist two nonadjacent vertices in S ; otherwise μ ( S ) : = + ∞ . [S. Fujita, Degree conditions for the partition of a graph into cycles, edges and isolated vertices, Discrete Math. 309 (2009) 3534–3540] proved that if μ ( V ( G ) ) ≥ ( n − k + 1 ) / 2 , then G contains k vertex-disjoint subgraphs H 1 , … , H k such that ⋃ i = 1 k V ( H i ) = V ( G ) and each H i is a cycle or K 2 or K 1 unless G is an exceptional graph. In this paper, we generalize this result as follows: if μ ( S ) ≥ ( n − k + 1 ) / 2 , then G contains k vertex-disjoint subgraphs H 1 , … , H k such that S ⊆ ⋃ i = 1 k V ( H i ) and each H i is a cycle or K 2 or K 1 unless G is an exceptional graph.