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.