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The groups of the generalized Petersen graphs

  • Roberto Frucht (a1), Jack E. Graver (a2) and Mark E. Watkins (a2)

1. Introduction. For integers n and k with 2 ≤ 2k < n, the generalized Petersen graph G(n, k) has been defined in (8) to have vertex-set

and edge-set E(G(n, k)) to consist of all edges of the form

where i is an integer. All subscripts in this paper are to be read modulo n, where the particular value of n will be clear from the context. Thus G(n, k) is always a trivalent graph of order 2n, and G(5, 2) is the well known Petersen graph. (The subclass of these graphs with n and k relatively prime was first considered by Coxeter ((2), p. 417ff.).)

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(1)Carmichael, R. D.Introduction to the theory of groups of finite order (Ginn, Boston, 1937).
(2)Coxeter, H. S. M.Self-dual configurations and regular graphs. Bull. Amer. Math. Soc. 56 (1950), 413455.
(3)Coxeter, H. S. M. and Moser, W. O. J.Generators and relations for discrete groups, Springer's Ergeb. N.F. 14 (Berlin, 1965).
(4)Foster, Ronald M.Census of trivalent symmetrical graphs, I, presented at the Second Waterloo Combinational Conference,University of Waterloo,Waterloo, Ontario, in April, 1966.
(5)Frucht, Roberto.Die gruppe des Petersen'schen Graphen und der Kantensysteme der regulären Polyeder. Comment. Math. Helv. 9, (1936/1937), 217223.
(6)Tutte, W. T.A family of cubical graphs. Proc. Cambridge Philos. Soc. 43 (1947), 459474.
(7)Tutte, W. T.Connectivity in graphs (University of Toronto Press, Toronto, 1966).
(8)Watkins, Mark E.A Theorem on Tait colorings with an application to the generalized Petersen graphs. J. Combinational Theory 6 (1969), 152164.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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