Extremal digraphs on Meyniel-type condition for hamiltonian cycles in balanced bipartite digraphs

TL;DR

This paper characterizes extremal balanced bipartite digraphs satisfying a Meyniel-type degree condition for Hamiltonian cycles and establishes a tight lower bound for traceability.

Contribution

It identifies the extremal structures for the Meyniel-type condition and extends the result to traceability with a tight lower bound.

Findings

Characterization of extremal digraphs under the condition d(u)+d(v) ≥ 3a

Proof that the bound 3a is tight for Hamiltonicity

Establishment of a tight bound 3a-1 for traceability

Abstract

Let $D$ be a strong balanced digraph on $2a$ vertices. Adamus et al. have proved that $D$ is hamiltonian if $d(u)+d(v)\ge 3a$ whenever $uv\notin A(D)$ and $vu\notin A(D)$. The lower bound $3a$ is tight. In this paper, we shall show that the extremal digraph on this condition is two classes of digraphs that can be clearly characterized. Moreover, we also show that if $d(u)+d(v)\geq 3a-1$ whenever $uv\notin A(D)$ and $vu\notin A(D)$, then $D$ is traceable. The lower bound $3a-1$ is tight.