Properly colored 2-factors of edge-colored complete bipartite graphs

Abstract

Let Gc be an edge-colored graph. The minimum color degree of Gc, denoted δc(G), is the largest integer k such that for every vertex v, there are at least k distinct colors on edges incident with v. A cycle in Gc is called properly colored if any of its adjacent edges have distinct colors. A properly colored 2-factor of Gc is a spanning subgraph consisting of vertex-disjoint properly colored cycles of Gc. It was conjectured in the literature that, if δc(Km,n)≥m+n4+1, then each vertex of Km,nc is contained in a properly colored cycle of length l for any even integer l with 4≤l≤min⁡{2m,2n}. In this paper, we consider the existence of properly colored 2-factor in edge-colored bipartite graphs and obtain the following two results: (i) if δc(Kn,n)>3n/4, then Kn,nc contains a properly colored 2-factor. (ii) For every number ε with 0<ε<1/2, there exists an integer n0(ε) such that every Kn,nc with n≥n0(ε) and δc(Kn,n)≥(1/2+ε)n contains a properly colored 2-factor.