On pre-Hamiltonian cycles in balanced bipartite digraphs

TL;DR

This paper investigates conditions under which strongly connected balanced bipartite digraphs contain long cycles, specifically focusing on the existence of pre-Hamiltonian cycles when the underlying undirected graph is not 2-connected.

Contribution

It establishes new degree conditions ensuring the presence of cycles of length 2a-2 in such digraphs, extending previous cycle existence results.

Findings

If the underlying undirected graph is not 2-connected and degree conditions are met, a cycle of length 2a-2 exists.

The exception is a specific digraph of order ten where the cycle does not exist.

Provides a characterization of when long cycles are guaranteed in balanced bipartite digraphs.

Abstract

Let $D$ be a strongly connected balanced bipartite directed graph of order $2a\geq 10$. 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: {\it If the underlying undirected graph of $D$ is not 2-connected and $max\{d(x), d(y)\}\geq 2a-2$ for every dominating pair of vertices $\{x,y\}$, then $D$ contains a cycle of length $2a-2$ unless $D$ is isomorphic to a certain digraph of order ten which we specify.