Abstract
Graph Theory We prove a sharp Meyniel-type criterion for hamiltonicity of a balanced bipartite digraph: For a≥2, a strongly connected balanced bipartite digraph D on 2a vertices is hamiltonian if d(u)+d(v)≥3a whenever uv∉A(D) and vu∉A(D). As a consequence, we obtain a sharp sufficient condition for hamiltonicity in terms of the minimal degree: a strongly connected balanced bipartite digraph D on 2a vertices is hamiltonian if δ(D)≥3a/2.
Highlights
1.1 ResultsThe main goal of this article is to prove a Meyniel-type sufficient condition for hamiltonicity of a balanced bipartite digraph
Our object of study in the present article is the class of bipartite digraphs satisfying the following Meyniel-type condition
For k ≥ 0, we will say that D satisfies condition (Mk) when d(u) + d(v) ≥ 3a + k for every pair of distinct vertices u, v ∈ V (D) such that uv ∈/ A(D) and vu ∈/ A(D)
Summary
The main goal of this article is to prove a Meyniel-type sufficient condition for hamiltonicity of a balanced bipartite digraph. The second example (due to Amar and Manoussakis [2]) shows that, for every a ≥ 3 and every 1 ≤ l < a/2, there is a non-hamiltonian strongly connected balanced bipartite digraph D(a, l) on 2a vertices with δ(D(a, l)) = a + l. Notice that conditions (Mk) cannot be weakened to apply only to pairs of vertices from the opposite colour classes (ala Theorem 1.8) This follows from the fact that there exist strongly connected non-hamiltonian balanced bipartite tournaments (Example 1.14 below). T (a, l) is strongly connected and vacuously satisfies condition (M1) ( condition (M0)) for every pair of vertices from the opposite colour classes, but T (a, l) contains no hamiltonian cycle
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