Two vertex-disjoint cycles in a graph

LetG be a graph of ordern ≥ 6 with minimum degree at least ⌈(n + 1)/2⌉. Then, for any two integerss andt withs ≥ 3,t ≥ 3 ands + t ≤ n, G contains two vertex-disjoint cycles of lengthss andt, respectively, unless thatn, s andt are odd andG is isomorphic toK(n−1)/2,(n−1)/2 + K1. We also show that ifG is a graph of ordern ≥ 8 withn even and minimum degree at leastn/2, thenG contains two vertex-disjoint cycles with any given even lengths provided that the sum of the two lengths is at mostn.