On the Hamiltonian Number of a Plane Graph

The Hamiltonian number of a connected graph is the minimum of the lengths of the closed spanning walks in the graph. In 1968, Grinberg published a necessary condition for the existence of a Hamiltonian cycle in a plane graph, formulated in terms of the degrees of its faces. We show how Grinberg’s theorem can be adapted to provide a lower bound on the Hamiltonian number of a plane graph