Abstract
In the instructions accompanying William Hamilton’s Icosian Game, it was written (by Hamilton) that every five consecutive vertices on a dodecahedron can be extended to produce a round trip on the dodecahedron that visits each vertex exactly once. This led to concepts for Hamiltonian graphs G dealing with (1) for any ordered list of k vertices in G, there exists a Hamiltonian cycle in G encountering these k vertices (not necessarily consecutively) in the given order and (2) determining the largest positive integer k for which any ordered list of k consecutive vertices in G lies on some Hamiltonian cycle in G as a path of order k. Whether G is Hamiltonian or not, there is a cyclic ordering of the vertices of G the sum of whose distances of consecutive vertices is minimum. These ideas are discussed in this chapter along with open questions dealing with them.
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Acknowledgements
I am grateful to Professor Gary Chartrand for suggesting the concept of Hamiltonian extension to me and kindly providing useful information on this topic. I also thank Stephen Hedetniemi, one of editors of this book, for numerous valuable suggestions.
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Zhang, P. (2016). Hamiltonian Extension. In: Gera, R., Hedetniemi, S., Larson, C. (eds) Graph Theory. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-31940-7_2
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DOI: https://doi.org/10.1007/978-3-319-31940-7_2
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