Cycles of each even lengths in balanced bipartite digraphs

Abstract

Let $D$ be a strongly connected balanced bipartite directed graph of order $2a\geq 4$. Let $x,y$ be distinct vertices in $D$. $\{x,y\}$ dominates a vertex $z$ if $x\rightarrow z$ and $y\rightarrow z$; in this case, we call the pair $\{x,y\}$ dominating. In this paper we prove: (i). If $a\geq 4$ and $ max\{d(x), d(y)\}\geq 2a-1$ for every dominating pair of vertices $\{x,y\}$, $D$ then contains a cycle of length $2a-2$ or $D$ is a directed cycle. (ii). If $D$ contains a cycle of length $2a-2\geq 6$ and $max \{d(x), d(y)\}\geq 2a-2$ for every dominating pair of vertices $\{x,y\}$, then for any $k$, $1\leq k\leq a-1$, $D$ contains a cycle of length $2k$. (iii). If $a\geq 4$ and $ max\{d(x), d(y)\}\geq 2a-1$ for every dominating pair of vertices $\{x,y\}$, then for every $k$, $1\leq k\leq a$, $D$ contains a cycle of length $2k$ unless $D$ is isomorphic to only one exceptional digraph of order eight.