Abstract: For an integer t, where t ≥ 2, let σ_t(G) denote the minimum degree sum of an independent set with t vertices in a graph G. We prove that for two integers k, t with k ≥ 3, t ≥ 4, every graph G with |V(G)| ≥ kt + 1.5k + t and σ_t(G) ≥ 2kt -t +1 contains k disjoint cycles. This result improves the theorem of Ma and Yan by optimizing the lower bound of |V(G)|. In addition, we also improve the theorem of Gould et al. for t = 4 and k ≥ 2.

Key words: Graph, Degree sum, Disjoint cycles