A sufficient condition for a balanced bipartite digraph to be hamiltonian

TL;DR

This paper establishes a new sufficient condition involving dominating pairs and degree bounds for strong balanced bipartite digraphs to contain Hamiltonian cycles, with the bounds proven to be sharp.

Contribution

It introduces a novel degree condition based on dominating pairs that guarantees Hamiltonicity in strong balanced bipartite digraphs, extending previous criteria.

Findings

The condition is sufficient for Hamiltonicity in strong balanced bipartite digraphs.

The degree bounds provided are proven to be sharp.

The paper offers a new perspective on Hamiltonian cycle criteria in bipartite digraphs.

Abstract

We describe a new type of sufficient condition for a balanced bipartite digraph to be hamiltonian. Let $D$ be a balanced bipartite digraph and $x,y$ 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 that a strong balanced bipartite digraph $D$ on $2a$ vertices contains a hamiltonian cycle if, for every dominating pair of vertices $\{x, y\}$, either $d(x)\ge 2a-1$ and $d(y)\ge a+1$ or $d(x)\ge a+1$ and $d(y)\ge 2a-1$. The lower bound in the result is sharp.