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Bicirculants via Imprimitivity Block Systems

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Abstract

A graph-theoretic environment is used to study the connection between imprimitivity and semiregularity, two concepts arising naturally in the context of permutation groups. Among other, it is shown that a connected arc-transitive graph admitting a nontrivial automorphism with two orbits of odd length, together with an imprimitivity block system consisting of blocks of size 2, orthogonal to these two orbits, is either the canonical double cover of an arc-transitive circulant or the wreath product of an arc-transitive circulant with the empty graph \({\bar{K}}_2\) on two vertices.

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References

  1. Alspach, B., Conder, M.D.E., Marušič, D., Xu, M.Y.: A classification of 2-arc-transitive circulants. J. Algebr. Combin. 5, 83–86 (1996)

    Article  MathSciNet  Google Scholar 

  2. Antončič, I., Hujdurović, A., Kutnar, K.: A classification of pentavalent arc-transitive bicirculants. J. Algebr. Combin. 41, 643–668 (2015)

    Article  MathSciNet  Google Scholar 

  3. Arroyo, A., Hubard, I., Kutnar, K., O’Reilly, E., Šparl, P.: Classification of symmetric Tabačjn graphs. Graphs Combin. 31, 1137–1153 (2015)

    Article  MathSciNet  Google Scholar 

  4. Cameron, P. J. (ed.) Problems from the fifteenth British combinatorial conference. Discrete Math. 167/168, 605–615 (1997)

  5. Devillers, A., Giudici, M., Jin, W.: Arc-transitive bicirculants. arXiv:1904.06467

  6. Dobson, T., Miklavič, Š., Šparl, P.: On automorphism groups of deleted wreath products. Mediterr. J. Math. 16, 149 (2019). https://doi.org/10.1007/s00009-019-1422-y

    Article  MathSciNet  MATH  Google Scholar 

  7. Dobson, T., Morris, J.: Automorphism groups of wreath product digraphs. Electron. J. Combin. 16, \(\# P.17\), 1–30 (2009)

  8. Du, S., Malnič, A., Marušič, D.: Classification of 2-arc-transitive dihedrants. J. Combin. Theory Ser. B 98, 1349–1372 (2008)

    Article  MathSciNet  Google Scholar 

  9. Frucht, R., Graver, J.E., Watkins, M.E.: The groups of the generalized Petersen graphs. Proc. Camb. Philos. Soc. 70, 211–218 (1971)

    Article  MathSciNet  Google Scholar 

  10. Godsil, C.D.: On the full automorphism groups of a graph. Combinatorica 1, 243–256 (1981)

    Article  MathSciNet  Google Scholar 

  11. Jajcay, R., Miklavič, Š., Šparl, P., Vasiljević, G.: On certain edge-transitive bicirculants. Electron. J. Combin. 26, Paper No. 2.6 (2019)

  12. Kovács, I.: Classifying arc-transitive circulants. J. Algebr. Combin. 20, 353–358 (2004)

    Article  MathSciNet  Google Scholar 

  13. Kovács, I., Kutnar, K., Marušič, D.: Classification of edge-transitive rose window graphs. J. Graph Theory 65, 216–231 (2010)

    Article  MathSciNet  Google Scholar 

  14. Kovács, I., Kuzman, B., Malnič, A., Wilson, S.: Characterization of edge-transitive 4-valent bicirculants. J. Graph Theory 69, 441–463 (2012)

    Article  MathSciNet  Google Scholar 

  15. Kovács, I., Kuzman, B., Malnič, A.: On non-normal arc transitive 4-valent dihedrants. Acta Math. Sin. Engl. Ser. 26, 1485–1498 (2010)

    Article  MathSciNet  Google Scholar 

  16. Li, C.H.: Permutation groups with a cyclic regular subgroup and arc transitive circulants. J. Algebr. Combin. 21, 131–136 (2005)

    Article  MathSciNet  Google Scholar 

  17. Malnič, A., Nedela, R., Škoviera, M.: Lifting graph automorphisms by voltage assignments. Eur. J. Combin. 21, 927–947 (2000)

    Article  MathSciNet  Google Scholar 

  18. Marušič, D.: On vertex symmetric digraphs. Discrete Math. 36, 69–81 (1981)

    Article  MathSciNet  Google Scholar 

  19. Marušič, D.: On 2-arc-transitivity of Cayley graphs. J. Combin. Theory Ser. B 87, 162–196 (2003)

  20. Marušič, D.: Corrigendum to: On 2-arc-transitivity of Cayley graphs [J. Combin. Theory Ser. B 87(1), 162–196 (2003)]. J. Combin. Theory Ser. B 96, 761–764 (2006)

  21. Marušič, D., Pisanski, T.: Symmetries of hexagonal molecular graphs on the torus. Croat. Chem. Acta 73, 969–981 (2000)

    Google Scholar 

  22. Pisanski, T.: A classification of cubic bicirculants. Discrete Math. 307, 567–578 (2007)

    Article  MathSciNet  Google Scholar 

  23. Pisanski, T.: Not every bipartite double cover is canonical. Bull. ICA 82, 51–55 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Sabidussi, G.: The lexicographic product of graphs. Duke Math. J. 28, 573–578 (1961)

    Article  MathSciNet  Google Scholar 

  25. Wilson, S.: Rose window graphs. Ars Math. Contemp. 1, 7–18 (2008)

    Article  MathSciNet  Google Scholar 

  26. Xu, M.Y.: Automorphism groups and isomorphisms of Cayley digraphs. Discrete Math. 182, 309–320 (1998)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author wishes to thank Ademir Hujdurović and Klavdija Kutnar for helpful conversations about the material of this paper.

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  1. University of Primorska, UP IAM, Muzejski trg 2, 6000, Koper, Slovenia

    Dragan Marušič

  2. Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000, Ljubljana, Slovenia

    Dragan Marušič

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  1. Dragan Marušič
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Correspondence to Dragan Marušič.

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This work is supported in part by the Slovenian Research Agency (I0-0035, research program P1-0285 and research projects N1-0062, J1-9108, J1-1695, N1-0140 and J1-2451)

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Marušič, D. Bicirculants via Imprimitivity Block Systems. Mediterr. J. Math. 18, 116 (2021). https://doi.org/10.1007/s00009-021-01771-z

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  • DOI: https://doi.org/10.1007/s00009-021-01771-z

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