A sufficient condition for pre-Hamiltonian cycles in bipartite digraphs

Abstract

Let $D$ be a strongly connected balanced bipartite directed graph of order $2a\ge 10$ other than a directed cycle. Let $x,y$ be distinct vertices in $D$. $\{x,y\}$ dominates a vertex $z$ if $x\to z$ and $y\to z$; in this case, we call the pair $\{x,y\}$ dominating. In this paper we prove: If $max\{d(x),d(y)\}\ge 2a-2$ for every dominating pair of vertices $\{x,y\}$, then $D$ contains cycles of all lengths $2,4,\dots ,2a-2$ or $D$ is isomorphic to a certain digraph of order ten which we specify.

15 pages