International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 30, Pages 1613-1616
Hamiltonian paths on Platonic graphs
Department of Mathematics, Saint Peter's College, Jersey City 07306, NJ, USA
Received 13 July 2003
Copyright © 2004 Brian Hopkins. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We develop a combinatorial method to show that the dodecahedron
graph has, up to rotation and reflection, a unique Hamiltonian
cycle. Platonic graphs with this property are called
topologically uniquely Hamiltonian. The same method is used to
demonstrate topologically distinct Hamiltonian cycles on the
icosahedron graph and to show that a regular graph embeddable on
the -holed torus is topologically uniquely Hamiltonian.