Thomas M. Lewis
Published 2015-08-27Version 1
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.
Lower bound for the rank of rigidity matrix of 4-valent graphs under various connectivity assumptions
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