Hamiltonian Paths and Cycles in Planar Graphs

We examine the problem of counting the number of Hamiltonian paths and Hamiltonian cycles in outerplanar graphs and planar graphs, respectively. We give an O(nαn) upper bound and an Ω(αn) lower bound on the maximum number of Hamiltonian paths in an outerplanar graph with n vertices, where α ≈ 1.46557 is the unique real root of α = α + 1. For any positive integer n ≥ 6, we define an outerplanar graph G, called a ZigZag outerplanar graph, such that the number of Hamiltonian paths starting at a single vertex in G is the maximum over all possible outerplanar graphs with n vertices. Finally, we prove a 2.2134n upper bound on the number of Hamiltonian cycles in planar graphs, which improves the previously best known upper bound 2.3404n.