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

In an edge -colored graph G, let dmon(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 dmon(v) over all vertices v E V(G) is called the maximum monochromatic -degree of G, denoted by triangle mon(G). Li et al. in 2019 conjectured that every edge -colored complete graph G of order n with triangle mon(G) <= n - 3k + 1 contains k vertexdisjoint 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 triangle mon(G) <= n - 2k contains k PC cycles of different lengths. They further got the condition triangle mon(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. (c) 2024 Elsevier B.V. All rights reserved.