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

Abstract

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 $u\in V\left(K_{n,n}^c\right)$ and for every even integer k with $4\le k\le 2n$, there exists a PC k-cycle containing u. The minimum color degree ${\delta }^c\left(K_{n,n}^c\right)$ 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 $t\ge 3$, every $K_{n,n}^c$ with ${\delta }^c\left(K_{n,n}^c\right)\ge \frac{2n}{3}+t$ 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 $\epsilon >0$, there exists an integer $n_0\left(\epsilon \right)$ such that every $K_{n,n}^c$ with $n\ge n_0\left(\epsilon \right)$ is properly vertex-even-pancyclic, provided that ${\delta }^c\left(K_{n,n}^c\right)\ge (\frac{2}{3}+\epsilon )n$.