Geodesic
Finite edge-transitive oriented graphs of valency four: a global approach
Jehan A. Al-bar1 ; Ahmad N. Al-kenani1 ; Najat M. Muthana1 ; Cheryl E. Praeger2 ; Pablo Spiga3
1King Abdulaziz University
2University of Western Australia (also affiliated with King Abdulaziz University)
3University of Milano-Bicocca
The electronic journal of combinatorics, Tome 23 (2016) no. 1
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We develop a new framework for analysing finite connected, oriented graphs of valency four, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of `basic' graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient. We anticipate that each of these additional properties will facilitate effective further analysis, and we demonstrate that this is so for the quasiprimitive basic graphs. Here we obtain strong restrictions on the group involved, and construct several infinite families of such graphs which, to our knowledge, are different from any recorded in the literature so far. Several open problems are posed in the paper.
DOI : 10.37236/4779
Classification : 05C25, 20B25
Mots-clés : oriented graph, graph quotient, transitive group, quasiprimitive permutation group

Jehan A. Al-bar  1   ; Ahmad N. Al-kenani  1   ; Najat M. Muthana  1   ; Cheryl E. Praeger  2   ; Pablo Spiga  3

1 King Abdulaziz University
2 University of Western Australia (also affiliated with King Abdulaziz University)
3 University of Milano-Bicocca 
@article{10_37236_4779,
     author = {Jehan A. Al-bar and Ahmad N. Al-kenani and Najat M. Muthana and Cheryl E. Praeger and Pablo Spiga},
     title = {Finite edge-transitive oriented graphs of valency four: a global approach},
     journal = {The electronic journal of combinatorics},
     year = {2016},
     volume = {23},
     number = {1},
     doi = {10.37236/4779},
     zbl = {1329.05142},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/4779/}
}
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Jehan A. Al-bar; Ahmad N. Al-kenani; Najat M. Muthana; Cheryl E. Praeger; Pablo Spiga. Finite edge-transitive oriented graphs of valency four: a global approach. The electronic journal of combinatorics, Tome 23 (2016) no. 1. doi: 10.37236/4779

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