Sufficient conditions for Hamiltonian cycles in bipartite digraphs

TL;DR

This paper establishes new sharp sufficient conditions for the existence of Hamiltonian cycles in balanced bipartite directed graphs, improving previous theorems and confirming a longstanding conjecture for large graphs.

Contribution

It provides two novel sharp conditions involving dominating pairs that guarantee Hamiltonian cycles in balanced bipartite digraphs, refining earlier results and resolving a conjecture.

Findings

First condition improves Wang's theorem for certain bipartite digraphs.

Second condition confirms a conjecture by Bang-Jensen, Gutin, and Li for large graphs.

Identifies an exceptional non-Hamiltonian digraph of order eight.

Abstract

We prove two sharp sufficient conditions for hamiltonian cycles in balanced bipartite directed graph. Let $D$ be a strongly connected balanced bipartite directed graph of order $2a$. Let $x,y$ be distinct vertices in $D$. $\{x,y\}$ dominates a vertex $z$ if $x\rightarrow z$ and $y\rightarrow z$; in this case, we call the pair $\{x,y\}$ dominating. (i) {\it If $a\geq 4$ and $max \{d(x), d(y)\}\geq 2a-1$ for every dominating pair of vertices $\{x,y\}$, then either $D$ is hamiltonian or $D$ is isomorphic to one exceptional digraph of order eight.} (ii) {\it If $a\geq 5$ and $d(x)+d(y)\geq 4a-3$ for every dominating pair of vertices $\{x,y\}$, then $D$ is hamiltonian.} The first result improves a theorem of R. Wang (arXiv:1506.07949 [math.CO]), the second result, in particular, establishes a conjecture due to Bang-Jensen, Gutin and Li (J. Graph Theory , 22(2), 1996) for strongly connected balanced bipartite digraphs of order at least ten.