A digraph D is k-transitive if the existence of a directed path (v0, v1, . . . , vk), of length k implies that (v0, vk) ∈ A(D). Clearly, a 2-transitive digraph is a transitive digraph in the usual sense. Transitive digraphs have been characterized as compositions of complete digraphs on an acyclic transitive digraph. Also, strong 3 and 4-transitive digraphs have been characterized. In this work we analyze the structure of strong k-transitive digraphs having a cycle of length at least k. We show that in most cases, such digraphs are complete digraphs or cycle extensions. Also, the obtained results are used to prove some particular cases of the Laborde-Payan-Xuong Conjecture.
@article{bwmeta1.element.doi-10_7151_dmgt_1765,
author = {C\'esar Hern\'andez-Cruz and Juan Jos\'e Montellano-Ballesteros},
title = {Some Remarks On The Structure Of Strong K-Transitive Digraphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {34},
year = {2014},
pages = {651-671},
zbl = {1303.05075},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1765}
}
César Hernández-Cruz; Juan José Montellano-Ballesteros. Some Remarks On The Structure Of Strong K-Transitive Digraphs. Discussiones Mathematicae Graph Theory, Tome 34 (2014) pp. 651-671. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1765/
[1] J. Bang-Jensen and G. Gutin, Digraphs. Theory, Algorithms and Applications (Springer-Verlag Berlin, Heidelberg New York, 2002). | Zbl 1001.05002
[2] R. Diestel, Graph Theory 3rd Edition (Springer-Verlag Berlin, Heidelberg New York, 2005).
[3] H. Galeana-Sánchez and C. Hernández-Cruz, k-kernels in k-transitive and k-quasi- transitive digraphs, Discrete Math. 312 (2012) 2522-2530. doi:10.1016/j.disc.2012.05.005
[4] A. Ghouila-Houri, Caract´erisation des graphes non orient´es dont on peut orienterles arrˆetes de mani`ere `a obtenir le graphe d’une relation d’ordre, C. R. Acad. Sci. Paris 254 (1962) 1370-1371.
[5] C. Hernández-Cruz, 3-transitive digraphs, Discuss. Math. Graph Theory 32 (2012) 205-219. doi:10.7151/dmgt.1613
[6] C. Hernández-Cruz, 4-transitive digraphs I: The structure of strong 4-transitive di- graphs, Discuss. Math. Graph Theory 33 (2013) 247-260. doi:10.7151/dmgt.1645 | Zbl 1293.05136
[7] J.M. Laborde, C. Payan and N.H. Xuong, Independent sets and longest directed paths in digraphs, in: Graphs and other Combinatorial Topics, Prague, M. Fiedler (Ed(s)), (Teubner, Leipzig, 1983) 173-177. | Zbl 0528.05034
[8] R. Wang, A conjecture on k-transitive digraphs, Discrete Math. 312 (2012) 1458-1460. doi:0.1016/j.disc.2012.01.011 | Zbl 1237.05092
[9] R. Wang and S. Wang, Underlying graphs of 3-quasi-transitive digraphs and 3- transitive digraphs, Discuss. Math. Graph Theory 33 (2013) 429-436. doi:10.7151/dmgt.1680 | Zbl 1293.05141