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On groups all of whose Haar graphs are Cayley graphs

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

Abstract

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.

Keywords

Haar graph Cayley graph Vertex-transitive graph 

Mathematics Subject Classification

05E18 (Primary) 20B25 (Secondary) 

Notes

Acknowledgements

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

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

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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