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Polygonal Graphs

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Horizons of Combinatorics

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 17))

Abstract

A near-polygonal graph is a graph г with a distinguished set C of cycles of common length m such that each path of length two lies in a unique element of C. If m is the girth of г then the graph is called polygonal. We describe various constructions of polygonal and near-polygonal graphs, and some attempts toward their classification.

Supported in part by the NSF and the NSA.

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References

  1. D. Archdeacon, Densely embedded graphs, J. Comb. Theory B, 54 (1992), 13–36.

    Article  MATH  MathSciNet  Google Scholar 

  2. D. Archdeacon and M. Perkel, Constructing polygonal graphs of large girth and degree, Congr. Num., 70 (1990), 81–85.

    MathSciNet  Google Scholar 

  3. M. Brown and R. Connelly, On graphs with a constant link, in: New Directions in the Theory of Graphs (F. Harary, ed.), Academic Press (New York, 1973), pp. 19–51.

    Google Scholar 

  4. P. J. Cameron, Suborbits in transitive permutation groups, in: Combinatorial Group Theory, Math. Centre Tracts, 57 (1974), pp. 98–129.

    Google Scholar 

  5. L. H. Clark, R. C. Entringer, J. E. McCanna and L. A. Székely, Extremal problems for local properties of graphs, Australasian J. Combin., 4 (1991), 25–31.

    MATH  Google Scholar 

  6. H. S. M. Coxeter and W. O. J. Moser, Generators and Relations Discrete Groups, volume 14 of Ergeb. Math. Grenzgeb., Springer-Verlag (Berlin, Göttingen, Heidelberg, 4th edition, 1957).

    Google Scholar 

  7. The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.4 (Aachen-St. Andrews, 2004).

    Google Scholar 

  8. M. Giudici, C. H. Li, C. E. Praeger, Á. Seress and V. I. Trofimov, On limit graphs of finite vertex-primitive graphs, J. Comb. Theory A, 114 (2007), 110–134.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. L. Gross and T. W. Tucker, Topological Graph Theory, John Wiley & Sons (New York, 1987).

    MATH  Google Scholar 

  10. A. A. Ivanov, On 2-transitive graphs of girth 5, European J. Comb., 8 (1987), 393–420.

    MATH  Google Scholar 

  11. C. H. Li and Á. Seress, Symmetrical path-cycle covers of a graph and polygonal graphs, J. Comb. Theory A, 114 (2007), 35–51.

    Article  MATH  MathSciNet  Google Scholar 

  12. S. Negami, Uniqueniss and faithfulness of embeddings of graphs into surfaces, Ph. Thesis, Tokyo Inst. of Technology (1985).

    Google Scholar 

  13. M. Perkel, Ph. D. Thesis, Univ. Michigan (1977).

    Google Scholar 

  14. M. Perkel, Bounding the valency of polygonal graphs with odd girth, Canad. J. Math., 31 (1979), 1307–1321.

    MATH  MathSciNet  Google Scholar 

  15. M. Perkel, A characterization of J 1 in terms of its geometry, Geom. Dedicata, 9 (1980), 291–298.

    Article  MathSciNet  Google Scholar 

  16. M. Perkel, Polygonal graphs of valency four, Congr. Num., 35 (1982), 387–400.

    MathSciNet  Google Scholar 

  17. M. Perkel, Trivalent polygonal graphs, Congr. Num., 45 (1984), 45–70.

    MathSciNet  Google Scholar 

  18. M. Perkel, Trivalent polygonal graphs of girth 6 and 7, Congr. Num., 49 (1985), 129–138.

    MathSciNet  Google Scholar 

  19. M. Perkel, Near-polygonal graphs, Ars Comb., 26A (1988), 149–170.

    MathSciNet  Google Scholar 

  20. M. Perkel and C. E. Praeger. On narrow hexagonal graphs with a 3-homogeneous suborbit, J. Algebraic Comb., 13 (2001), 257–273.

    Article  MATH  MathSciNet  Google Scholar 

  21. Á. Seress, Toward the classification of s-arc transitive graphs, in: Groups St. Andrews 2005, (C. Campbell, E. Robertson, ed.), volume 340 of London Math. Soc. Lecture Note Series, Cambridge Univ. Press (2007), pp. 401–414.

    Google Scholar 

  22. Á. Seress and T. Szabó, Dense graphs with cycle neighborhoods, J. Comb. Theory B, 63 (1995), 281–293.

    Article  MATH  Google Scholar 

  23. S. Stahl and A. T. White, Genus embeddings for some complete tripartite graphs, Discrete Math., 14 (1976), 279–296.

    Article  MATH  MathSciNet  Google Scholar 

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© 2008 János Bolyai Mathematical Society and Springer-Verlag

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Seress, Á. (2008). Polygonal Graphs. In: Győri, E., Katona, G.O.H., Lovász, L., Sági, G. (eds) Horizons of Combinatorics. Bolyai Society Mathematical Studies, vol 17. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77200-2_9

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