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

Abstract

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∈V(G) is called the maximum monochromatic-degree of G, denoted by Δmon(G). Li et al. in 2019 conjectured that every edge-colored complete graph G of order n with Δmon(G)≤n−3k+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 Δmon(G)≤n−2k contains k PC cycles of different lengths. They further got the condition Δmon(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.