On the Hamiltonian Number of a Planar Graph

TL;DR

This paper adapts Grinberg's theorem to establish a lower bound on the Hamiltonian number of planar graphs, advancing understanding of spanning walks in such graphs.

Contribution

It introduces a novel application of Grinberg's theorem to estimate the Hamiltonian number in planar graphs, providing a new analytical tool.

Findings

Provides a lower bound on the Hamiltonian number for planar graphs

Adapts Grinberg's theorem for this purpose

Enhances theoretical understanding of Hamiltonian properties in planar graphs

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.