Cycles of each even lengths in balanced bipartite digraphs (1607.04074v1)

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.