Discussiones Mathematicae Graph Theory 33(2) (2013) 429-435
doi: 10.7151/dmgt.1680

[BIBTex] [PDF] [PS]

Underlying graphs of 3-quasi-transitive digraphs and 3-transitive digraphs

Ruixia Wang and Shiying Wang

School of Mathematical Sciences,
Shanxi University,
Taiyuan, Shanxi, 030006, PR China


A digraph is 3-quasi-transitive (resp. 3-transitive), if for any path x0x1x2x3 of length 3, x0 and x3 are adjacent (resp. x0 dominates x3). César Hernández-Cruz conjectured that if D is a 3-quasi-transitive digraph, then the underlying graph of D, UG(D), admits a 3-transitive orientation. In this paper, we shall prove that the conjecture is true.

Keywords: graph orientation, 3-quasi-transitive digraph, 3-transitive digraph

2010 Mathematics Subject Classification: 05C20.


[1]J. Bang-Jensen, Kings in quasi-transitive digraphs, Discrete Math. 185 (1998) 19--27, doi: 10.1016/S0012-365X(97)00179-9.
[2]J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Applications ( Springer, London, 2000).
[3]C. Hernández-Cruz, 3-transitive digraphs, Discuss. Math. Graph Theory 32 (2012) 205--219, doi: 10.7151/dmgt.1613.
[4]A. Ghouila-Houri, Caractérization des graphes non orientés dont onpeut orienter les arrêtes de maniére àobtenir le graphe dune relation dordre, Comptes Rendus de l?Académie des Sciences Paris 254 (1962) 1370?-1371.
[5]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.
[6]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.
[7]S. Wang and R. Wang, Independent sets and non-augmentable paths in arc-locally in-semicomplete digraphs and quasi-arc-transitive digraphs, Discrete Math. 311 (2011) 282--288, doi: 10.1016/j.disc.2010.11.009.

Received 5 November 2011
Revised 15 May 2012
Accepted 20 June 2012