Journal of the
Korean Mathematical Society JKMS

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 $e_1,\ldots, e_k$ 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 $C_1,\ldots, C_k$ of orders $n_1,\ldots ,n_k$, 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.

Keywords: cycles, disjoint cycles, cycle coverings

MSC numbers: 05C38, 05C70, 05C75