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

Consider a complete balanced bipartite graph $K_{n,n}$ and let $K^c_{n,n}$ 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 $K^c_{n,n}$ 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 $u\in V(K_{n,n}^c)$ and for every even integer $k$ with $4 \leq k \leq 2n$, there exists a PC $k$-cycle containing $u$. The minimum color degree $\delta^c(K^c_{n,n})$ of $K^c_{n,n}$ 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 $t\geq 3$, every $K^c_{n,n}$ with $\delta^c(K^c_{n,n})\geq \frac{2n}{3}+t$ 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 $\varepsilon>0$, there exists an integer $n_0(\varepsilon)$ such that every $K^c_{n,n}$ with $n\geq n_0(\varepsilon)$ is properly vertex-even-pancyclic, provided that $\delta^c(K^c_{n,n})\geq (\frac{2}{3}+\varepsilon)n$.