On the maximum number of cycles in a planar graph

  • Authors:

  • R. E. L. Aldred;Carsten Thomassen

  • Affiliations:

  • Department of Mathematics and Statistics, University of Otago P. O. Box 56, Dunedin, New Zealand;Department of Mathematics, Technical University of Denmark DK-2800 Lyngby, Denmark

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2008

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Abstract

Let G be a graph on p vertices with q edges and let r = q - p = 1. We show that G has at most ${15\over 16} 2^{r}$ cycles. We also show that if G is planar, then G has at most 2r - 1 = o(2r - 1) cycles. The planar result is best possible in the sense that any prism, that is, the Cartesian product of a cycle and a path with one edge, has more than 2r - 1 cycles. © Wiley Periodicals, Inc. J. Graph Theory 57: 255–264, 2008