A degree sum condition for hamiltonicity in balanced bipartite digraphs

TL;DR

This paper establishes a new degree sum condition that guarantees Hamiltonian cycles in strongly connected balanced bipartite digraphs, expanding understanding of conditions for Hamiltonicity in directed graphs.

Contribution

It introduces a novel degree sum criterion involving pairs of vertices with common neighbors that ensures Hamiltonicity in balanced bipartite digraphs.

Findings

Strongly connected balanced bipartite digraphs with degree sum ≥ 3a are Hamiltonian.

The condition applies for all pairs sharing a common in- or out-neighbor.

The result generalizes previous Hamiltonicity conditions in bipartite digraphs.

Abstract

We prove that a strongly connected balanced bipartite digraph $D$ of order $2a$ is hamiltonian, provided $a\geq3$ and $d(x)+d(y)\geq 3a$ for every pair of vertices $x$, $y$ with a common in-neighbour or a common out-neighbour in $D$.