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Journal of Geometry

, 110:15 | Cite as

One-factorisations of complete graphs arising from hyperbolae in the Desarguesian affine plane

  • Nicola PaceEmail author
  • Angelo Sonnino
Article
  • 16 Downloads

Abstract

In a recent paper Korchmáros et al. (J Combin Theory Ser A 160:62–83, 2018) the geometry of finite planes is exploited for the construction of one-factorisations of the complete graph \(K_n\) from configurations of points in \(\mathrm {PG}(2,q)\). Here we provide an alternative procedure where the vertices of \(K_n\) correspond to the points of a hyperbola in \(\mathrm {AG}(2,q)\). In this way, we obtain one-factorisations for parameters which are either not covered by the constructions in Korchmáros et al. (J Combin Theory Ser A 160:62–83, 2018), or isomorphic to known examples but arising from different geometric configurations.

Keywords

Complete graph one-factorisation projective plane affine plane 

Mathematics Subject Classification

Primary 05C70 Secondary 51E21 

Notes

Acknowledgements

This research was carried out within the activities of the GNSAGA—Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni of the Italian INdAM. Nicola Pace was supported by the Alexander von Humboldt Foundation with funds from the German Federal Ministry of Education and Research (BMBF).

References

  1. 1.
    Bonisoli, A., Labbate, D.: One-factorizations of complete graphs with vertex-regular automorphism groups. J. Combin. Des. 10(1), 1–16 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bonisoli, A., Rinaldi, G.: Quaternionic starters. Graphs Combin. 21(2), 187–195 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bosma, W., Cannon, J.J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Buratti, M.: Abelian 1-factorization of the complete graph. Eur. J. Combin. 22(3), 291–295 (2001)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cameron, P.J., Korchmáros, G.: One-factorizations of complete graphs with a doubly transitive automorphism group. Bull. Lond. Math. Soc. 25(1), 1–6 (1993)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dinitz, J.H., Stinson, D.R.: Some new perfect one-factorizations from starters in finite fields. J. Graph Theory 13(4), 405–415 (1989)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hartman, A., Rosa, A.: Cyclic one-factorization of the complete graph. Eur. J. Combin. 6(1), 45–48 (1985)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hirschfeld, J.W.P.: Projective Geometries Over Finite Fields, second edn. Oxford Mathematical Monographs. The ClarendonPress, Oxford University Press, New York (1998)Google Scholar
  9. 9.
    Kiss, Gy.: One-factorizations of complete multigraphs and quadrics in PG (n, q). J. Combin. Des. 10(2), 139–143 (2002)Google Scholar
  10. 10.
    Kiss, Gy., Pace, N., Sonnino, A.: On circular-linear one-factorizations of the complete graph \(K_n\). Preprint (2018)Google Scholar
  11. 11.
    Kiss, Gy., Rubio-Montiel, C.: A note on m-factorizations of complete multigraphs arising from designs. ARS Math. Contemp. 8(1), 163–175 (2015)Google Scholar
  12. 12.
    Korchmáros, G.: Cyclic one-factorization with an invariant one-factor of the complete graph. ARS Combin. 27, 133–138 (1989)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Korchmáros, G.: Sharply transitive 1-factorizations of the complete graph with an invariant 1-factor. J. Combin. Des. 2(4), 185–196 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Korchmáros, G., Pace, N., Sonnino, A.: One-factorisations of complete graphs arising from ovals in finite planes. J. Combin. Theory Ser. A 160, 62–83 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Korchmáros, G., Siciliano, A., Sonnino, A.: 1-factorizations of complete multigraphs arising from finite geometry. J. Combin. Theory Ser. A 93(2), 385–390 (2001)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mendelsohn, E., Rosa, A.: One-factorizations of the complete graph—a survey. J. Graph Theory 9(1), 43–65 (1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Pace, N., Sonnino, A.: One-factorisations of complete graphs constructed in Desarguesian planes of odd square order. Preprint (2018)Google Scholar
  18. 18.
    Pasotti, A., Pellegrini, M.A.: Symmetric 1-factorizations of the complete graph. Eur. J. Combin. 31(5), 1410–1418 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Rinaldi, G.: Nilpotent 1-factorizations of the complete graph. J. Combin. Des. 13(6), 393–405 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sonnino, A.: One-factorizations of complete multigraphs arising from maximal \((k;n)\) -arcs in \({\rm PG}(2,2^h)\). Discrete Math. 231(1–3), 447–451 (2001)Google Scholar
  21. 21.
    Wallis, W.D.: One-Factorizations of Complete Graphs, Contemporary Design Theory, Wile-Interscience Series in Discrete Mathematics and Optimization, pp. 593–631. Wiley, New York (1992)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Chair of Operations ResearchTechnical University of MunichMunichGermany
  2. 2.Dipartimento di Matematica, Informatica ed EconomiaUniversità degli Studi della BasilicataPotenzaItaly

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