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 n simple cycles and at least 2.0845 n Hamilton cycles. Based on counting arguments for perfect matchings we prove that 2.3404 n 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 n simple cycles. This reduces the previous gap between the upper and lower bound for the exponential growth from 1.03 to 0.46