aequationes mathematicae

, Volume 23, Issue 1, pp 80–97 | Cite as

Enumeration of hamiltonian paths in Cayley diagrams

  • David Housman
Research Paper


LetG be a group generated by a subset of elementsS. The Cayley diagram ofG givenS is the labeled directed graph with vertices identified with the elements ofG and (v, u) is an edge labeledh ifhS anduh=v. The sequence of elements ofS corresponding to the edges transversed in a hamiltonian path (whose initial vertex is the identity) is called a group generating sequence (abbreviatedggs) inS.

In this paper a minimal upper bound for the number ofggs's in a pair of generator elements for any two-generated group is given. For all groups of the formG=〈a, b:b n =1,a m =b r ,ba=ab−1〉 wherem is even, it is shown that the number ofggs's in {a, b} is 1+m(n−1)/2. An algorithm is developed that yields the number ofggs's for two-generated groupsG=〈a, b〉 for which 〈ba−1〉⊲G. Explicit forms for the countedggs's are also provided.

AMS (1980) subject classification

Primary 05C25 Secondary 05C30 


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  1. [1]
    Curran, S.,Hamiltonian paths and circuits in Cayley diagrams of abelian groups. Preliminary report.Google Scholar
  2. [2]
    Gallian, J. A.,Group theory and the design of a letter facing machine. Amer. Math. Monthly84 (1977), 285–287.MathSciNetGoogle Scholar
  3. [3]
    Gallian, J. A. andMarttila, C. A.,Reorienting regular n-gons. Aequationes Math.20 (1980), 97–103.Google Scholar
  4. [4]
    Holsztyński, W. andStrube, R. F. E.,Paths and circuits in finite groups. Discrete Math.22 (1978), 263–272.CrossRefGoogle Scholar
  5. [5]
    Keating, K.,The conjunction of two Cayley diagrams. Discrete Math. (to appear).Google Scholar
  6. [6]
    Klerlein, J. B.,Hamiltonian cycles in Cayley color graphs. J. Graph Theory2 (1978), 65–68.Google Scholar
  7. [7]
    Klerlein, J. B. andStarling, G.,Hamiltonian cycles in Cayley color graphs of semi-direct products. In:Proc. 9th S-E Conf. Combinatorics, Graph Theory, and Computing, 1978, pp. 411–435.Google Scholar
  8. [8]
    Letzter, G.,Hamiltonian circuits in Cartesian products with a metacyclic factor. Preprint.Google Scholar
  9. [9]
    Nathanson, M. B.,Partial products in finite groups. Discrete Math.15 (1976), 201–203.CrossRefGoogle Scholar
  10. [10]
    Nijenhuis, A. andWilf, H. S.,Combinatorial Algorithms. Academic Press, New York, 1975.Google Scholar
  11. [11]
    Rankin, R. A.,A campanological problem in group theory. Proc. Cambridge Philos. Soc.44 (1948), 17–25.Google Scholar
  12. [12]
    Rankin, R. A.,A campanological problem in group theory, II Proc. Cambridge Philos. Soc.62 (1966), 11–18.Google Scholar
  13. [13]
    Trotter, W. T. andErdös, P.,When the Cartesian product of directed cycles is hamiltonian. J. Graph Theory2 (1978), 137–142.Google Scholar
  14. [14]
    Witte, D. S.,On hamiltonian circuits in Cayley diagrams. Discrete Math.38 (1982), 99–108.CrossRefGoogle Scholar
  15. [15]
    Witte, D., Letzter, G., andGallian, J.,Hamiltonian circuits in Cartesian products of Cayley diagrams. Preprint.Google Scholar

Copyright information

© Birkhäuser Verlag 1981

Authors and Affiliations

  • David Housman
    • 1
  1. 1.Center for Applied MathematicsCornell UniversityIthacaU.S.A.

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