Degree sum conditions on two disjoint cycles in graphs

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Abstract Let G be a simple undirected graph on n vertices. A set of subgraphs of G is disjoint if no two of them have any common vertex in G. Suppose that n 1 , n 2 are two integers with n 1 , n 2 ≥ 3 and n = n 1 + n 2 . We prove that if d ( x ) + d ( y ) ≥ n + 4 for any pair of vertices x , y of G with x y ∉ E ( G ) , then G contains two disjoint cycles of length n 1 and n 2 .