2017 MATRIX Annals pp 337-341 | Cite as

# Biquasiprimitive Oriented Graphs of Valency Four

## Abstract

In this short note we describe a recently initiated research programme aiming to use a normal quotient reduction to analyse finite connected, oriented graphs of valency 4, admitting a vertex- and edge-transitive group of automorphisms which preserves the edge orientation. In the first article on this topic (Al-bar et al. Electr J Combin 23, 2016), a subfamily of these graphs was identified as ‘basic’ in the sense that all graphs in this family are normal covers of at least one ‘basic’ member. These basic members can be further divided into three types: quasiprimitive, biquasiprimitive and cycle type. The first and third of these types have been analysed in some detail. Recently, we have begun an analysis of the basic graphs of biquasiprimitive type. We describe our approach and mention some early results. This work is on-going. It began at the Tutte Memorial MATRIX Workshop.

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## Notes

### Acknowledgements

Both authors are grateful for the opportunity to participate in the Tutte Memorial MATRIX retreat which gave them to chance to commence work on this problem. The authors also thank Georgina Liversidge for some useful discussions at the retreat. The first author acknowledges support of a Research Training Program Scholarship at the University of Melbourne. The second author is grateful for project funding from the Deanship of Scientific Research, King Abdulaziz University (grant no. HiCi/H1433/363-1) which provided the opportunity to focus on this research problem.

## References

- 1.Al-bar, J.A., Al-kenani, A.N., Muthana, N.M., Praeger, C.E., Spiga, P.: Finite edge-transitive oriented graphs of valency four: a global approach. Electron. J. Combin.
**23**(1), #P1.10 (2016). arXiv: 1507.02170Google Scholar - 2.Al-Bar, J.A., Al-Kenani, A.N., Muthana, N.M., Praeger, C.E.: Finite edge-transitive oriented graphs of valency four with cyclic normal quotients. J. Algebraic Combin.
**46**(1), 109–133 (2017)MathSciNetCrossRefGoogle Scholar - 3.Al-bar, J.A., Al-kenani, A.N., Muthana, N.M., Praeger, C.E.: A normal quotient analysis for some families of oriented four-valent graphs. Ars Mat. Contemp.
**12**(2), 361–381 (2017)MathSciNetCrossRefGoogle Scholar - 4.Marušič, D.: Half-transitive group actions on finite graphs of valency 4. J. Combin. Theory B
**73**(1), 41–76 (1998)MathSciNetCrossRefGoogle Scholar - 5.Marušič, D., Praeger, C.E.: Tetravalent graphs admitting half-transitive group actions: alternating cycles. J. Combin. Theory B
**75**(2), 188–205 (1999)MathSciNetCrossRefGoogle Scholar - 6.Marušič, D., Šparl, P.: On quartic half-arc-transitive metacirculants. J. Algebraic Combin.
**28**, 365–395 (2008)MathSciNetCrossRefGoogle Scholar - 7.Morris, J., Praeger, C.E., Spiga, P.: Strongly regular edge-transitive graphs. Ars Mat. Contemp.
**2**(2), 137–155 (2009)MathSciNetCrossRefGoogle Scholar - 8.Praeger, C.E.: An O’Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs. J. Lond. Math. Soc.
**2**(2), 227–239 (1993)MathSciNetCrossRefGoogle Scholar - 9.Praeger, C.E.: Finite normal edge-transitive Cayley graphs. Bull. Aust. Math. Soc.
**60**(2), 207–220 (1999)MathSciNetCrossRefGoogle Scholar - 10.Praeger, C.E.: Finite transitive permutation groups and bipartite vertex-transitive graphs. Ill. J. Math.
**47**(1–2), 461–475 (2003)MathSciNetzbMATHGoogle Scholar