Vertex-transitive Haar graphs that are not Cayley graphs

摘要

In a recent paper (arXiv:1505.01475 ) Est\'elyi and Pisanski raised aquestion whether there exist vertex-transitive Haar graphs that are not Cayleygraphs. In this note we construct an infinite family of trivalent Haar graphsthat are vertex-transitive but non-Cayley. The smallest example has 40 verticesand is the well-known Kronecker cover over the dodecahedron graph $G(10,2)$,occurring as the graph $40$ in the Foster census of connected symmetrictrivalent graphs.