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Let $k \ge 3$ be an integer, $H_{k}(G)$ be the set of vertices of degree atleast $2k$ in a graph $G$, and $L_{k}(G)$ be the set of vertices of degree atmost $2k-2$ in $G$. In 1963, Dirac and Erd\H{o}s proved that $G$ contains $k$(vertex-)disjoint cycles whenever $|H_{k}(G)| - |L_{k}(G)| \ge k^{2} + 2k - 4$.The main result of this paper is that for $k \ge 2$, every graph $G$ with$|V(G)| \ge 3k$ containing at most $t$ disjoint triangles and with $|H_{k}(G)|- |L_{k}(G)| \ge 2k + t$ contains $k$ disjoint cycles. This yields that if $k\ge 2$ and $|H_{k}(G)| - |L_{k}(G)| \ge 3k$, then $G$ contains $k$ disjointcycles. This generalizes the Corr\'{a}di-Hajnal Theorem, which states thatevery graph $G$ with $H_{k}(G) = V(G)$ and $|H_{k}(G)| \ge 3k$ contains $k$disjoint cycles.

Strengthening theorems of Dirac and Erd\H{o}s on disjoint cycles