On a Problem of Wang Concerning the Hamiltonicity of Bipartite Digraphs

TL;DR

This paper investigates a problem posed by Wang on the Hamiltonicity of bipartite digraphs, proving that under certain degree conditions, such graphs contain a cycle factor and have pairs of vertices sharing a common out-neighbour.

Contribution

The paper establishes that bipartite digraphs satisfying Wang's degree conditions always contain a cycle factor and pairs of vertices with a common out-neighbour, advancing understanding of Hamiltonicity criteria.

Findings

Contains a cycle factor

Every vertex has a partner with a common out-neighbour

Satisfies Wang's degree conditions

Abstract

R. Wang (Discrete Mathematics and Theoretical Computer Science, vol. 19(3), 2017) proposed the following problem. \textbf{Problem.} Let $D$ be a strongly connected balanced bipartite directed graph of order $2a\geq 8$. Suppose that $d(x)\geq 2a-k$, $ d(y)\geq a+k$ or $d(y)\geq 2a-k$, $ d(x)\geq a+k$ for every pair of vertices $x,y$ with a common out-neighbour, where $2 \leq k\leq a/2$. Is $D$ Hamiltonian? In this paper, we prove that if a digraph $D$ satisfies the conditions of this problem, then (i) $D$ contains a cycle factor, (ii) for every vertex $x\in V(D)$ there exists a vertex $y\in V(D)$ such that $x$ and $y$ have a common out-neighbour.