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A dominating pair condition for a balanced bipartite digraph to be Hamiltonian. (English) Zbl 1447.05093

In a digraph, a vertex \(x\) is said to dominate vertex \(y\) if there is an arc from \(x\) to \(y\). Similarly, a vertex \(z\) is said to dominate vertices \(x\) and \(y\) if \(z\) dominates both \(x\) and \(y\). Likewise, the pair of vertices \(x\) and \(y\) dominates \(z\) if both \(x\) and \(y\) dominate \(z\). [the first author, Discrete Math. Theor. Comput. Sci. 19, No. 3, Paper No. 11, 12 p. (2017; Zbl 1401.05177)] gave a dominating pair sufficient condition for Hamiltonicity of a balanced bipartite digraph. In the present paper, it is shown that a strong balanced bipartite digraph of order \(2a\) contains a Hamiltonian cycle if for every dominating pair of vertices \(x\) and \(y\), their degrees \(d(x)\) and \(d(y)\) satisfy certain degree inequalities involving \(a\).

MSC:

05C20 Directed graphs (digraphs), tournaments
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
05C45 Eulerian and Hamiltonian graphs

Citations:

Zbl 1401.05177
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References:

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