Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



An infinite graph of girth $ 12$

Author: Asia Ivić Weiss
Journal: Trans. Amer. Math. Soc. 283 (1984), 575-588
MSC: Primary 52A45; Secondary 05C25, 20F05, 51M20
MathSciNet review: 737885
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: From the regular hyperbolic honeycomb $ \{ 3,6,3\} $ we derive regular honeycombs with finite numbers of toroidal cells. Joining centers of faces of these honeycombs to the midpoints of its edges we obtain trivalent symmetrical graphs. We investigate the relation between these honeycombs, their groups and the graphs embedded in them.

References [Enhancements On Off] (What's this?)

  • [1891] L. Bianchi, Geometrische Darstellung der Gruppen linearer Substitutional mit ganzen complexen Coefficienten nebst Anwendungen auf die Zahlentheorie, Math. Ann. 38, 313-333. MR 1510677
  • [1968] N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris. MR 0240238 (39:1590)
  • [1939] H. S. M. Coxeter, The abstract groups $ {G^{m,n,p}}$, Trans. Amer. Math. Soc. 45, 73-150. MR 1501984
  • [1968] -, Twelve geometric essays, Southern Illinois Univ. Press, Carbondale, Ill. MR 0310745 (46:9843)
  • [1973] -, Regular polytopes, 3rd ed., Dover, New York. MR 0370327 (51:6554)
  • [1974] -, Regular complex polytopes, Cambridge Univ. Press, London and New York. MR 0370328 (51:6555)
  • [1980] -, The edges and faces of a $ 4$-dimensional polytope, Congress. Numer. 28, 309-334. MR 608436 (82i:51030)
  • [1981] H. S. M. Coxeter, R. Frucht and D. L. Powers, Zero-symmetric graphs: Trivalent graphical regular representations of groups, Academic Press, New York. MR 658666 (83h:05043)
  • [1980] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 4th ed., Springer, Berlin. MR 562913 (81a:20001)
  • [1950] H. S. M. Coxeter and G. T. Whitrow, World structure and non-Euclidean honeycombs, Proc. Roy. Soc. London Ser. A 201, 417-437. MR 0041576 (12:866e)
  • [1966] R. M. Foster, A census of trivalent symmetrical graphs. I, presented at the Conference on Graph Theory and Combinatorial Analysis, University of Waterloo, Ontario.
  • [1964] C. W. L. Garner, Polyhedra and honeycombs in hyperbolic space, Ph.D. Thesis, University of Toronto.
  • [1976] B. Grünbaum, Regularity of graphs, complexes and designs, Colloq. Internat. CNRS, Problèmes Combinatoires et Théorie des Graphes, Orsay.
  • [1929] D. M. Y. Sommerville, An introduction to the geometry of $ n$ dimensions, Methuen, London.
  • [1966] W. T. Tutte, Connectivity in graphs, Univ. of Toronto Press. MR 0210617 (35:1503)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 52A45, 05C25, 20F05, 51M20

Retrieve articles in all journals with MSC: 52A45, 05C25, 20F05, 51M20

Additional Information

Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society