On the Hamiltonian Number of a Planar Graph

Abstract

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 planar graph, formulated in terms of the lengths of its face cycles. We show how Grinberg's theorem can be adapted to provide a lower bound on the Hamiltonian number of a planar graph.