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Imbeddings of metacyclic cayley graphs

Part of the Lecture Notes in Mathematics book series (LNM,volume 642)

Abstract

Any group whose commutator subgroup and commutator quotient group are both cyclic is called metacyclic. The Cayley graph corresponding to any generating set for a metacyclic group is called a metacyclic Cayley graph. Whereas the earliest work on Cayley graph imbeddings concentrates mainly on planarity, recent work of A.T. White and others shows that higher genus imbeddings are accessible, especially for abelian groups. Since metacyclic groups are an especially tractable kind of nonabelian groups, this paper beings a development of an imbedding theory for nonabelian Cayley graphs by considering the metacyclic Cayley graphs. The author has previously used voltage graphs (the duals of the current graphs of Gustin, Ringel, and Youngs) to show that a special subclass of metacyclic groups has toroidal Cayley graphs. In this paper, toroidal Cayley graphs are constructed for two additional subclasses of metacyclic groups.

The author is an Alfred P. Sloan Fellow. His research was partially supported by NSF Contract MPS74-05481-A01 at Columbia University.

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References

  1. J.W. Alexander, Note on Riemann spaces, Bull. Amer. Math. Soc. 26 (1920), 370–372.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. S.R. Alpert and J.L. Gross, Components of branched coverings of a current graph, J. Combinatorial Theory, to appear in 1976.

    Google Scholar 

  3. L. Babai, Chromatic number and subgraphs of Cayley graphs, These Conference Proceedings.

    Google Scholar 

  4. H.S.M. Coxeter and W.O.J. Moser, Generators and Relations for Discrete Groups, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 14 (Springer, Berlin, 1972).

    Google Scholar 

  5. J.L. Gross, Voltage graphs, Discrete Math. 9 (1974), 239–246.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. J.L. Gross, The genus of nearly complete graphs — Case 6, Aequationes Math. 13 (1975), 243–249.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. J.L. Gross, Every connected regular graph of even degree is a Schreier coset graph, J. Combinatorial Theory, to appear.

    Google Scholar 

  8. J.L. Gross and S.R. Alpert, The topological theory of current graphs, J. Combinatorial Theory 17 (1974), 218–233.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. J.L. Gross and T.W. Tucker, Quotients of complete graphs: revisiting the Heawood map-coloring problem, Pacific J. Math. 55 (1974), 391–402.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. J.L. Gross and T.W. Tucker, Generating all graph coverings by permutation voltage assignments, to appear.

    Google Scholar 

  11. W. Gustin, Orientable embeddings of Cayley graphs, Bull. Amer. Math. Soc. 69 (1963), 272–275.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. P. Himelwright, On the genus of Hamiltonian groups, Specialist Thesis, Western Michigan Univ. 1972.

    Google Scholar 

  13. M. Jungerman and A.T. White, On the genus of finite abelian groups, Abstract No. 9, Graph Theory Newsletter May, 1976.

    Google Scholar 

  14. H. Levinson, On the genera of graphs of group presentations, Ann. New York Acad. Sci. 175 (1970), 277–284.

    MathSciNet  MATH  Google Scholar 

  15. W. Maschke, The representation of finite groups, Amer. J. Math. 18 (1896), 156–194.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. G. Ringel, Map Color Theorem, Springer, New York, 1974.

    CrossRef  MATH  Google Scholar 

  17. S. Stahl, Self-dual embeddings of graphs, Ph.D. Dissertation, Western Michigan Univ. 1975.

    Google Scholar 

  18. C. M. Terry, L.R. Welch, and J.W.T. Youngs, The genus of K12s, J. Combinatorial Theory 2 (1967), 43–60.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. A.T. White, On the genus of a group, Trans. Amer. Math. Soc. 173 (1972), 203–214.

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. A.T. White, Graphs, Groups and Surfaces, North-Holland, Amsterdam, 1973.

    MATH  Google Scholar 

  21. H. Zassehaus, The Theory of Groups (2nd ed.), Chelsea, New York, 1958.

    Google Scholar 

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© 1978 Springer-Verlag Berlin Heidelberg

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Gross, J.L. (1978). Imbeddings of metacyclic cayley graphs. In: Alavi, Y., Lick, D.R. (eds) Theory and Applications of Graphs. Lecture Notes in Mathematics, vol 642. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0070377

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  • DOI: https://doi.org/10.1007/BFb0070377

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