On groups all of whose Haar graphs are Cayley graphs

  • Yan-Quan FengEmail author
  • István Kovács
  • Da-Wei Yang


A Cayley graph of a group H is a finite simple graph \(\Gamma \) such that \(\mathrm{Aut}(\Gamma )\) contains a subgroup isomorphic to H acting regularly on \(V(\Gamma ),\) while a Haar graph of H is a finite simple bipartite graph \(\Sigma \) such that \(\mathrm{Aut}(\Sigma )\) contains a subgroup isomorphic to H acting semiregularly on \(V(\Sigma )\) and the H-orbits are equal to the bipartite sets of \(\Sigma \). A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that \(D_6, \, D_8, \, D_{10}\) and \(Q_8\) are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs. (A group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian.) This result will then be used to derive that every non-solvable group admits a non-Cayley Haar graph.


Haar graph Cayley graph Vertex-transitive graph 

Mathematics Subject Classification

05E18 (Primary) 20B25 (Secondary) 



The authors are greatly indebted to Tomaž Pisanski for his useful suggestions and improvements.


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Authors and Affiliations

  1. 1.Department of MathematicsBeijing Jiaotong UniversityBeijingChina
  2. 2.IAMUniversity of PrimorskaKoperSlovenia
  3. 3.FAMNITUniversity of PrimorskaKoperSlovenia
  4. 4.School of SciencesBeijing University of Posts and TelecommunicationsBeijingChina

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