Vertex-Transitive Haar Graphs That Are Not Cayley Graphs

  • Marston D. E. Conder
  • István Estélyi
  • Tomaž Pisanski
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 234)


In a recent paper in Electron. J. Combin. 23 (2016), Estélyi and Pisanski raised a question whether there exist vertex-transitive Haar graphs that are not Cayley graphs. In this note we construct an infinite family of trivalent Haar graphs that are vertex-transitive but non-Cayley. The smallest example has 40 vertices and is the well-known Kronecker cover over the dodecahedron graph G(10, 2), occurring as the graph ‘40’ in the Foster census of connected symmetric trivalent graphs.


Haar graph Cayley graph Vertex-transitive graph 

MSC 2010

05E18 (primary) 20B25 (secondary) 



We acknowledge with gratitude the use of the Magma system [3] for helping to find and analyse examples. We would also like to thank Klavdija Kutnar and István Kovács for fruitful conversations and for pointing out several crucial references, and the referees for helpful comments, and in particular, one of the referees for suggesting the family of 2-arc-regular covers of the Pappus graph as another potential source of examples.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Marston D. E. Conder
    • 1
  • István Estélyi
    • 2
    • 3
    • 4
  • Tomaž Pisanski
    • 3
    • 5
  1. 1.Mathematics DepartmentUniversity of AucklandAucklandNew Zealand
  2. 2.FMFUniversity of LjubljanaLjubljanaSlovenia
  3. 3.IAMUniversity of PrimorskaKoperSlovenia
  4. 4.NTISUniversity of West BohemiaPlzeň 3Czech Republic
  5. 5.FAMNITUniversity of PrimorskaKoperSlovenia

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