DISJOINT CYCLES WITH PRESCRIBED LENGTHS AND INDEPENDENT EDGES IN GRAPHS

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.