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Isomorphic factorisations VI: Automorphisms

Contributed Papers

Part of the Lecture Notes in Mathematics book series (LNM,volume 748)

Abstract

An isomorphic factorisation of a graph G is a partition of its line set E(G) into isomorphic subgraphs called factor graphs. The question investigated here is how those automorphisms of a complete graph K p which preserve an isomorphic factorisation can act in permuting the factor graphs. The group of these permutations is called the symmetry group. It is clear that the symmetric group S 2 is the only such factor group of degree 2. However, it is shown that all four permutation groups of degree 3 arise from isomorphic factorisations of complete graphs. For E 1 ×S 2 the smallest example requires 10 points. A natural conjecture is that every permutation group of degree d>2 is the symmetry group of an isomorphic factorisation of some complete graph. However no such representation is known for S 6, and it is shown that the representations of S 5 present certain irregularities.

Keywords

  • Symmetry Group
  • Complete Graph
  • Permutation Group
  • Factor Graph
  • Ramsey Number

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References

  1. B.A. Anderson, Symmetry groups of some perfect 1-factorizations of complete graphs, Discrete Math., 18 (1977), 227–234.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. B.A. Anderson, M.M. Barge and D. Morse, A recursive construction of asymmetric 1-factorizations, Aequationes Math., 15 (1977), 201–211.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. P.J. Cameron, On groups of degree n and n−1, and highly-symmetric edge colourings, J. London Math. Soc. (2), 9 (1975), 385–391.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. P.J. Cameron, Parallelisms of Complete Designs, (Cambridge Univ. Press, 1976).

    Google Scholar 

  5. P. Dembowski, Finite Geometries. (Springer-Verlag, New York, 1968).

    CrossRef  MATH  Google Scholar 

  6. L.E. Dickson and F.H. Safford, Solution to problem 8 (group theory), Amer. Math. Monthly, 13 (1906), 150–151.

    CrossRef  MathSciNet  Google Scholar 

  7. E.N. Gelling and R.E. Odeh, On 1-factorizations of the complete graph and relationships to round robin schedules, Proc. 3rd Manitoba Conf. on Numerical Math. (Utilitas, Winnipeg, 1974), 213–221.

    MATH  Google Scholar 

  8. F. Harary, Graph Theory. (Addison-Wesley, Reading, Mass., 1969).

    MATH  Google Scholar 

  9. F. Harary and R.W. Robinson, Generalised Ramsey theory IX: isomorphic factorizations IV: isomorphic Ramsey numbers, Pacific J. Math., to appear.

    Google Scholar 

  10. F. Harary, R.W. Robinson and N.C. Wormald, Isomorphic factorisations I: complete graphs, Trans. Amer. Math. Soc., 242 (1978), 243–260.

    MathSciNet  MATH  Google Scholar 

  11. F. Harary, R.W. Robinson and N.C. Wormald, Isomorphic factorisations III: complete multipartite graphs, Combinatorial Mathematics, Proceedings of the International Conference on Combinatorial Theory (Canberra), (Lecture Notes No. 686, Springer-Verlag, Berlin, 1978), 47–54.

    CrossRef  Google Scholar 

  12. F. Harary, R.W. Robinson and N.C. Wormald, Isomorphic factorisations V: directed graphs, Mathematika, to appear.

    Google Scholar 

  13. F. Harary and W.D. Wallis, Isomorphic factorizations II: combinatorial designs, Proc. 8th Southeastern Conf. on Combinatorics, Graph Theory and Computing. (Utilitas, Winnipeg, 1977), 13–28.

    MATH  Google Scholar 

  14. C.C. Lindner, E. Mendelsohn and A. Rosa, On the number of 1-factorizations of the complete graph, J. Combinatorial Theory (B), 20 (1976), 265–282.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Š. Porubský, Factorial regular representation of groups in complete graphs, to appear.

    Google Scholar 

  16. A.P. Street and W.D. Wallis, Sum-free sets, coloured graphs and designs, J. Austral. Math. Soc., 22 (A) (1976), 35–53.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. W.D. Wallis, A.P. Street and J.S. Wallis, Combinatorics: Room Squares, Sumfree Sets, Hadamard Matrices. (Lecture Notes No. 292, Springer-Verlag, Berlin, 1972).

    CrossRef  MATH  Google Scholar 

  18. B. Zelinka, Decomposition of the complete graph according to a given group, Mat. Časopis Sloven. Akad. Vied, 17 (1967), 234–239. (Czech, with English summary).

    MathSciNet  MATH  Google Scholar 

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© 1979 Springer-Verlag

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Robinson, R.W. (1979). Isomorphic factorisations VI: Automorphisms. In: Horadam, A.F., Wallis, W.D. (eds) Combinatorial Mathematics VI. Lecture Notes in Mathematics, vol 748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0102691

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  • DOI: https://doi.org/10.1007/BFb0102691

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