We conjecture that if $k{geq}2$ is an integer and G is a graph of order n with minimum degree at least (n+2k)/2, then for any k independent edges $e_1$, ${cdots}$, $e_k$ in G and for any integer partition $n=n_1+{cdots}+n_k$ with $n_i{geq}4(1{leq}i{leq}k)$, G has k disjoint cycles $C_1$, ${cdots}$, $C_k$ of orders $n_1$, ${cdots}$, $n_k$, respectively, such that $C_i$ passes through $e_i$ for all $1{leq}i{leq}k$. We show that this conjecture is true for the case k = 2. The minimum degree condition is sharp in general.