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On the Full Automorphism Group of a Hamiltonian Cycle System of Odd Order

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Abstract

It is shown that a necessary condition for an abstract group G to be the full automorphism group of a Hamiltonian cycle system is that G has odd order or it is either binary, or the affine linear group AGL(1, p) with p prime. We show that this condition is also sufficient except possibly for the class of non-solvable binary groups.

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References

  1. Akiyama, J., Kobayashi, M., Nakamura, G.: Symmetric Hamilton cycle decompositions of the complete graph. J. Comb. Des. 12, 39–45 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anderson, B.A., Ihrig, E.C.: Every finite solvable group with a unique element of order two, except the quaternion group, has a symmetric sequencing. J. Comb. Des. 1, 3–14 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bailey, R.A., Ollis, M.A., Preece, D.A.: Round-dance neighbour designs from terraces. Discret. Math. 266, 69–86 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bailey, R.A., Buratti, M., Rinaldi, G., Traetta, T.: On 2-pyramidal Hamiltonian cycle systems. Bull. Belg. Math. Soc. Simon Stevin 21, 747–758 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Biggs, N., White, A.T.: Permutation Groups and Combinatorial Structures. London Mathematical Society Lecture Note Series no. 33. Cambridge University Press, New York (1979)

    Book  Google Scholar 

  6. Bonisoli, A., Buratti, M., Mazzuoccolo, G.: Doubly transitive 2-factorizations. J. Comb. Des. 15, 120–132 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bonvicini, S., Mazzuoccolo, G., Rinaldi, G.: On 2-factorizations of the complete graph: from the \(k\)-pyramidal to the universal property. J. Comb. Des. 17, 211–228 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brualdi, R.A., Schroeder, M.W.: Symmetric Hamilton cycle decompositions of complete graphs minus a 1-factor. J. Comb. Des. 19, 1–15 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Buratti, M.:Sharply vertex-transitive Hamiltonian cycle systems of the complete and cocktail party graph, preprint

  10. Buratti, M., Capparelli, S., Merola, F., Rinaldi, G., Traetta, T.: A collection of results on Hamiltonian cycle systems with a nice automorphism group. Electron. Notes Discret. Math. 40C, 245–252 (2013)

    Article  Google Scholar 

  11. Buratti, M., Rinaldi, G., Traetta, T.: Some results on 1-rotational Hamiltonian cycle systems of the complete graph. J. Comb. Des. 22, 231–251 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Buratti, M., Del Fra, A.: Cyclic Hamiltonian cycle systems of the complete graph. Discret. Math. 279, 107–119 (2004)

    Article  MATH  Google Scholar 

  13. Buratti, M., Merola, F.: Dihedral Hamiltonian cycle systems of the cocktail party graph. J. Comb. Des. 21, 1–23 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Buratti, M., Merola, F.: Hamiltonian cycle systems that are both cyclic and symmetric. J. Comb. Des. 22, 367–390 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Buratti, M., Rinaldi, G.: 1-rotational \(k\)-factorizations of the complete graph and new solutions to the Oberwolfach problem. J. Comb. Des. 16, 87–100 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grannell, M.J., Griggs, T.S., Lovegrove, G.J.: Even-cycle systems with prescribed automorphism groups. J. Comb. Des. 21, 142–156 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jordon, H., Morris, J.: Cyclic Hamiltonian cycle systems of the complete graph minus a 1-factor. Discret. Math. 308, 2440–2449 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lovegrove, G.J.: Combinatorial designs and their automorphism groups. Ph.D. Thesis, The Open University (2008)

  19. Lovegrove, G.J.: Odd-cycle systems with prescribed automorphism groups. Discret. Math. 314, 6–13 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lucas, E.: Recreations Mathematiques. Gauthiers Villars, Paris (1892)

    Google Scholar 

  21. Mendelsohn, E.: Every group is the collineation group of some projective plane. J. Geom. 2, 97–106 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mendelsohn, E.: On the groups of automorphisms of Steiner triple and quadruple systems. J. Comb. Theory Ser. A 25, 97–104 (1978)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

Work of Marco Buratti and Tommaso Traetta performed under the auspicies of the G.N.S.A.G.A. of the C.N.R. (National Research Council) of Italy and supported by M.I.U.R. project “Disegni combinatorici, grafi e loro applicazioni, PRIN 2008”. Tommaso Traetta was supported by a fellowship of INdAM.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, via Vanvitelli 1, 06123, Perugia, Italy

    Marco Buratti & Tommaso Traetta

  2. Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK

    Graham J. Lovegrove

Authors
  1. Marco Buratti
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  2. Graham J. Lovegrove
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  3. Tommaso Traetta
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Correspondence to Tommaso Traetta.

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Buratti, M., Lovegrove, G.J. & Traetta, T. On the Full Automorphism Group of a Hamiltonian Cycle System of Odd Order. Graphs and Combinatorics 31, 1855–1865 (2015). https://doi.org/10.1007/s00373-015-1584-8

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  • DOI: https://doi.org/10.1007/s00373-015-1584-8

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