Vertex-Transitive Haar Graphs That Are Not Cayley Graphs
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.
KeywordsHaar graph Cayley graph Vertex-transitive graph
MSC 201005E18 (primary) 20B25 (secondary)
We acknowledge with gratitude the use of the Magma system  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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