# Minimum Reload Cost Cycle Cover in Complete Graphs

## TL;DR

This paper investigates the minimum reload cost cycle cover problem in complete graphs with specific 2-edge-colorings, proving the existence of zero-cost covers and providing polynomial-time algorithms for their construction.

## Contribution

It establishes conditions under which a zero reload cost cycle cover exists in complete graphs with equitable or nearly equitable 2-edge-colorings and offers efficient algorithms for constructing such covers.

## Key findings

- Zero reload cost cycle cover exists in most complete graphs with nearly equitable 2-edge-colorings.
- Polynomial-time algorithms are provided for constructing monochromatic cycle covers.
- The results hold except for small graphs with fewer than 13 vertices or specific special cases.

## Abstract

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.