Geodesic
Hamiltonian Cycle Problem in Strong k-Quasi-Transitive Digraphs with Large Diameter
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 2, pp. 685-690

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Let k be an integer with k ≥ 2. A digraph is k-quasi-transitive, if for any path x0x1... xk of length k, x0 and xk are adjacent. Let D be a strong k-quasi-transitive digraph with even k ≥ 4 and diameter at least k +2. It has been shown that D has a Hamiltonian path. However, the Hamiltonian cycle problem in D is still open. In this paper, we shall show that D may contain no Hamiltonian cycle with k ≥ 6 and give the sufficient condition for D to be Hamiltonian.
Keywords: quasi-transitive digraph, k -quasi-transitive digraph, Hamiltonian cycle 
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Wang, Ruixia. Hamiltonian Cycle Problem in Strong k-Quasi-Transitive Digraphs with Large Diameter. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 2, pp. 685-690. http://geodesic.mathdoc.fr/item/DMGT_2021_41_2_a20/