On the Number of Cycles in Planar Graphs

We investigate the maximum number of simple cycles and the maximum number of Hamiltonian cycles in a planar graph G with n vertices. Using the transfer matrix method we construct a family of graphs which have at least 2.4262 simple cycles and at least 2.0845 Hamilton cycles. Based on counting arguments for perfect matchings we prove that 2.3404 is an upper bound for the number of Hamiltonian cycles. Moreover, we obtain upper bounds for the number of simple cycles of a given length with a face coloring technique. Combining both, we show that there is no planar graph with more than 2.8927 simple cycles. This reduces the previous gap between the upper and lower bound for the exponential growth from 1.03 to 0.46.