Geodesic
On the Hamiltonian Number of a Plane Graph
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 1, pp. 171-181

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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.
Keywords: Hamiltonian cycle, Hamiltonian walk, Hamiltonian number, Hamiltonian spectrum, Grinberg’s theorem, planar graph 
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Lewis, Thomas M. On the Hamiltonian Number of a Plane Graph. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 1, pp. 171-181. http://geodesic.mathdoc.fr/item/DMGT_2019_39_1_a13/