A Note on Dominating Pair Degree Condition for Hamiltonian Cycles in Balanced Bipartite Digraphs

Abstract

Let D be a strong balanced bipartite digraph on 2a vertices. For $x,y,z\in V(D)$, if $x\rightarrow z$ and $y\rightarrow z$, then we call the pair $\{x,y\}$ dominating; if $z\rightarrow x$ and $z\rightarrow y$, then we call the pair $\{x,y\}$ dominated. In 2017, Adamus [Graphs and Combinatorics, 33(2017) 43–51] proved that if $d(x)+d(y)\ge 3a$ whenever $\{x,y\}$ is a dominating or dominated pair, then D is Hamiltonian. In 2021, Adamus [Discrete Mathematics, 344(3) (2021) 112240] proved that if only every dominating pair of vertices $\{x,y\}$ in D satisfies $d(x)+d(y)\ge 3a+1$, then D is Hamiltonian. In this paper, we show that the same conclusion is reached if we replace $3a+1$ with 3a in the above condition. The lower bound for 3a is sharp.