On the Meyniel condition for hamiltonicity in bipartite digraphs

TL;DR

This paper establishes a precise degree-based criterion for hamiltonicity in balanced bipartite digraphs, extending Meyniel's condition and providing sharp bounds for such graphs.

Contribution

It introduces a new sharp Meyniel-type condition for hamiltonicity in balanced bipartite digraphs, linking degree sums to Hamiltonian cycles.

Findings

A bipartite digraph with degree sum ≥ 3k+1 for non-adjacent pairs is hamiltonian.

Balanced bipartite digraphs with minimum degree ≥ (3k+1)/2 are hamiltonian.

The conditions are proven to be sharp and optimal.

Abstract

We prove a sharp Meyniel-type criterion for hamiltonicity of a balanced bipartite digraph: For k greater than or equal to 2, a bipartite digraph D with colour classes of cardinalities k is hamiltonian if the sum of degrees of vertices u and v is at least 3k+1 for every pair of vertices u, v such that D does not contain the arc uv nor vu. As a consequence, we obtain a sharp sufficient condition for hamiltonicity in terms of the minimal degree: a balanced bipartite digraph D on 2k vertices is hamiltonian if its minimal degree is at least (3k + 1)/2.