Properly colored even cycles in edge-colored complete balanced bipartite graphs

Consider a complete balanced bipartite graph K _n,n and let K _n,n ^c be an edge-colored version of K _n,n that is obtained from K _n,n by having each edge assigned a certain color. A subgraph H of K _n,n ^c is called properly colored (PC) if every two adjacent edges of H have distinct colors. K _n,n ^c is called properly vertex-even-pancyclic if for every vertex u∈V(K _n,n ^c) and for every even integer k with 4≤k≤2n, there exists a PC k-cycle containing u. The minimum color degree δ ^c(K _n,n ^c) of K _n,n ^c is the largest integer k such that for every vertex v, there are at least k distinct colors on the edges incident to v. In this paper we study the existence of PC even cycles in K _n,n ^c. We first show that, for every integer t≥3, every K _n,n ^c with δ ^c(K _n,n ^c)≥[Formula presented]+t contains a PC 2-factor H such that every cycle of H has a length of at least t. By using the probabilistic method and absorbing technique, we use the above result to further show that, for every ε>0, there exists an integer n ₀(ε) such that every K _n,n ^c with n≥n ₀(ε) is properly vertex-even-pancyclic, provided that δ ^c(K _n,n ^c)≥([Formula presented]+ε)n.