Wang, Ruixia Hamiltonian cycle problem in strong \(k\)-quasi-transitive digraphs with large diameter. (English) Zbl 1458.05095 Discuss. Math., Graph Theory 41, No. 2, 685-690 (2021). Summary: Let \(k\) be an integer with \(k \geq 2\). A digraph is \(k\)-quasi-transitive, if for any path \(x_0x_1\dots x_k\) of length \(k\), \(x_0\) and \(x_k\) are adjacent. Let \(D\) be a strong \(k\)-quasi-transitive digraph with even \(k \geq 4\) 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 \(k \geq 6\) and give the sufficient condition for \(D\) to be Hamiltonian. Cited in 1 Document MSC: 05C20 Directed graphs (digraphs), tournaments 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05C45 Eulerian and Hamiltonian graphs Keywords:quasi-transitive digraph; \(k\)-quasi-transitive digraph; Hamiltonian cycle PDFBibTeX XMLCite \textit{R. Wang}, Discuss. Math., Graph Theory 41, No. 2, 685--690 (2021; Zbl 1458.05095) Full Text: DOI References: [1] J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Applications (Springer-Verlag, London, 2000). · Zbl 1001.05002 [2] C. Hernández-Cruz and H. Galeana-Sánchez, k-kernels in k-transitive and k-quasi-transitive digraphs, Discrete Math. 312 (2012) 2522-2530. doi:10.1016/j.disc.2012.05.005 · Zbl 1246.05067 [3] H. Galeana-Sánchez. I.A. Goldfeder and I. Urrutia, On the structure of strong 3-quasi-transitive digraphs, Discrete Math. 310 (2010) 2495-2498. doi:10.1016/j.disc.2010.06.008 · Zbl 1213.05112 [4] H. Galeana-Sánchez, C. Hernández-Cruz and M.A. Juárez-Camacho, On the existence and number of (k +1) -kings in k-quasi-transitive digraphs, Discrete Math. 313 (2013) 2582-2591. doi:10.1016/j.disc.2013.08.007 · Zbl 1281.05068 [5] R. Wang and W. Meng, k-kings in k-quasitransitive digraphs, J. Graph Theory 79 (2015) 55-62. doi:10.1002/jgt.21814 · Zbl 1312.05066 [6] R. Wang, (k +1)-kernels and the number of k-kings in k-quasi-transitive digraphs, Discrete Math. 338 (2015) 114-121. doi:10.1016/j.disc.2014.08.009 · Zbl 1301.05152 [7] R. Wang and H. Zhang, Hamiltonian paths in k-quasi-transitive digraphs, Discrete Math. 339 (2016) 2094-2099. doi:10.1016/j.disc.2016.02.020 · Zbl 1336.05083 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.