Edge-disjoint properly colored cycles in edge-colored complete graphs.

In an edge-colored graph G , let d m o n (v) denote the maximum number of edges with the same color incident with a vertex v in G , called the monochromatic-degree of v. The maximum value of d m o n (v) over all vertices v ∈ V (G) is called the maximum monochromatic-degree of G , denoted by Δ m o n (G). Li et al. in 2019 conjectured that every edge-colored complete graph G of order n with Δ m o n (G) ≤ n − 3 k + 1 contains k vertex-disjoint properly colored (PC for short) cycles of length at most 4, and they showed that the conjecture holds for k = 2. Han et al. showed that every edge-colored complete graph G of order n with Δ m o n (G) ≤ n − 2 k contains k PC cycles of different lengths. They further got the condition Δ m o n (G) ≤ n − 6 for the existence of two vertex-disjoint PC cycles of different lengths. In this paper, we consider the problems of the existence of edge-disjoint PC cycles of length at most 4 (different lengths) in an edge-colored complete graph G of order n. [ABSTRACT FROM AUTHOR]