  The reload cost refers to the cost that occurs along a path on an
edge-colored graph when it traverses an internal vertex between two edges of
different colors. Galbiati et al.[1] introduced the Minimum Reload Cost Cycle
Cover problem, which is to find a set of vertex-disjoint cycles spanning all
vertices with minimum reload cost. They proved that this problem is strongly
NP-hard and not approximable within $1/\epsilon$ for any $\epsilon > 0$ even
when the number of colors is 2, the reload costs are symmetric and satisfy the
triangle inequality. In this paper, we study this problem in complete graphs
having equitable or nearly equitable 2-edge-colorings, which are edge-colorings
with two colors such that for each vertex $v \in V(G)$, $||c_1(v)| -|c_2(v)||
\leq 1$ or $||c_1(v)| -|c_2(v)|| \leq 2$, respectively, where $c_i(v)$ is the
set of edges with color $i$ that is incident to $v$. We prove that except
possibly on complete graphs with fewer than 13 vertices, the minimum reload
cost is zero on complete graphs with nearly equitable 2-edge-colorings by
proving the existence of a monochromatic cycle cover. Furthermore, we provide a
polynomial-time algorithm that constructs a monochromatic cycle cover in
complete graphs with an equitable 2-edge-coloring except possibly in a complete
graph with four vertices. Our algorithm also finds a monochromatic cycle cover
in complete graphs with a nearly equitable 2-edge-coloring except some special
cases.
