On the number of simple cycles in planar graphs

  • Helmut Alt
  • Ulrich Fuchs
  • Klaus Kriegel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1335)


Let C(G) denote the number of simple cycles of a graph G and let C(n) be the maximum of C(G) over all planar graphs with n nodes. We present a lower bound on C(n) constructing graphs with at least 2.27n cycles. Applying some probabilistic arguments we prove an upper bound of 3.37n.

We also discuss this question restricted to the subclasses of grid graphs, bipartite graphs, and of 3-colorable triangulated graphs.


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  1. 1.
    M. Blum, C. Hewitt, Automata on a 2-dimensional tape, 8th IEEE Conf. on SWAT, 1967, 155–160.Google Scholar
  2. 2.
    G. Ding, Bounding the Number of Circuits of a Graph, Combinatorica 16(3), 1996, 331–341.Google Scholar
  3. 3.
    D. Eppstein, Pesonal Communications, 1997Google Scholar
  4. 4.
    U. Fössmeier, M. Kaufmann, On Exact Solutions for the Rectilinear Steiner Problem, Part I: Theoretical Results, Technical Report, Computer Science Institute, Universität Tübingen, 1996.Google Scholar
  5. 5.
    P. J. Heawood, Map Color Theorem, Quarterly Journal of Pure and Applied Mathematics 24, 1890, 332–338.Google Scholar
  6. 6.
    T. R. Jensen, B. Toft, Graph Coloring Problems, Wiley, New York, 1995.Google Scholar
  7. 7.
    R. Steinberg, The State of the Three Color Problem, Annals of Discrete Mathematics 55, 1993, 211–248.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Helmut Alt
    • 1
  • Ulrich Fuchs
    • 1
  • Klaus Kriegel
    • 1
  1. 1.Institut für InformatikFreie Universität BerlinBerlin

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