# A sufficient condition for pre-Hamiltonian cycles in bipartite digraphs

## TL;DR

This paper establishes a sufficient degree condition for the existence of pre-Hamiltonian cycles in strongly connected balanced bipartite digraphs, identifying specific cycle lengths or a unique exceptional case.

## Contribution

It introduces a new degree condition involving dominating pairs that guarantees the presence of cycles of all even lengths up to a certain size or a specific exceptional digraph.

## Key findings

- If the maximum degree of vertices in dominating pairs is at least 2a-2, the digraph contains all even cycles up to length 2a-2.
- The result applies to strongly connected balanced bipartite digraphs of order at least 10.
- An exceptional digraph of order ten is characterized where the condition does not imply the cycle existence.

## Abstract

Let $D$ be a strongly connected balanced bipartite directed graph of order $2a\geq 10$ other than a directed cycle. 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: If $ max\{d(x), d(y)\}\geq 2a-2$ for every dominating pair of vertices $\{x,y\}$, then $D$ contains cycles of all lengths $2,4, \ldots , 2a-2$ or $D$ is isomorphic to a certain digraph of order ten which we specify.