A 2-factor with Short Cycles Passing Through Specified Independent Vertices in Graph

For a graph G, we define σ2(G) := min{d(u) + d(v)|u, v ≠ ∈ E(G), u ≠ v}. Let k ≥ 1 be an integer and G be a graph of order n ≥ 3k. We prove if σ2(G) ≥ n + k − 1, then for any set of k independent vertices v1,...,vk, G has k vertex-disjoint cycles C1,..., Ck of length at most four such that vi ∈ V(Ci) for all 1 ≤ i ≤ k. And show if σ2(G) ≥ n + k − 1, then for any set of k independent vertices v1,...,vk, G has k vertex-disjoint cycles C1,..., Ck such that vi ∈ V(Ci) for all 1 ≤ i ≤ k, V(C1) ∪...∪ V(Ck) = V(G), and |Ci| ≤ 4 for all 1 ≤ i ≤ k − 1.