A Meyniel-type condition for bipancyclicity in balanced bipartite digraphs

TL;DR

This paper establishes a Meyniel-type degree condition that guarantees bipancyclicity in strongly connected balanced bipartite digraphs, extending understanding of cycle structures in directed graphs.

Contribution

It introduces a new degree condition that ensures bipancyclicity or the presence of a Hamiltonian cycle in balanced bipartite digraphs.

Findings

The degree condition guarantees bipancyclicity or a directed cycle of length 2a.

The result applies to strongly connected balanced bipartite digraphs with order 2a.

The condition is tight for the class of graphs considered.

Abstract

We prove that a strongly connected balanced bipartite digraph $D$ of order $2a$, $a\geq3$, satisfying $d(u)+d(v)\geq 3a$ for every pair of vertices $u,v$ with a common in-neighbour or a common out-neighbour, is either bipancyclic or a directed cycle of length $2a$.