Four-Valent Oriented Graphs of Biquasiprimitive Type
Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 409-434.

Let 𝒪𝒢(4) denote the family of all graph-group pairs (Γ,G) where Γ is 4-valent, connected and G-oriented (G-half-arc-transitive). Using a novel application of the structure theorem for biquasiprimitive permutation groups of the second author, we produce a description of all pairs (Γ,G)𝒪𝒢(4) for which every nontrivial normal subgroup of G has at most two orbits on the vertices of Γ, and at least one normal subgroup has two orbits. In particular we show that G has a unique minimal normal subgroup N and that NT k for a simple group T and k{1,2,4,8}. This provides a crucial step towards a general description of the long-studied family 𝒪𝒢(4) in terms of a normal quotient reduction. We also give several methods for constructing pairs (Γ,G) of this type and provide many new infinite families of examples, covering each of the possible structures of the normal subgroup N.

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DOI: https://doi.org/10.5802/alco.161
Classification: 05C25,  20B25,  05E18
Keywords: Edge-transitive graphs, automorphism groups, oriented graphs, graph quotients, vertex-transitive graphs, quasiprimitive permutation groups, Cayley graphs.
@article{ALCO_2021__4_3_409_0,
     author = {Poznanovi\'c, Nemanja and Praeger, Cheryl E.},
     title = {Four-Valent {Oriented} {Graphs} of {Biquasiprimitive} {Type}},
     journal = {Algebraic Combinatorics},
     pages = {409--434},
     publisher = {MathOA foundation},
     volume = {4},
     number = {3},
     year = {2021},
     doi = {10.5802/alco.161},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.161/}
}
Poznanović, Nemanja; Praeger, Cheryl E. Four-Valent Oriented Graphs of Biquasiprimitive Type. Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 409-434. doi : 10.5802/alco.161. https://alco.centre-mersenne.org/articles/10.5802/alco.161/

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