On the maximum number of cycles in a planar graph
Robert E. L. Aldred, Carsten Thomassen · Journal of Graph Theory · 2007
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