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arXiv · 1701.06029

Hamiltonicity in locally finite graphs: two extensions and a counterexample

Karl Heuer
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Abstract

We state a sufficient condition for the square of a locally finite graph to contain a Hamilton circle, extending a result of Harary and Schwenk about finite graphs. We also give an alternative proof of an extension to locally finite graphs of the result of Chartrand and Harary that a finite graph not containing $K^4$ or $K_{2,3}$ as a minor is Hamiltonian if and only if it is $2$-connected. We show furthermore that, if a Hamilton circle exists in such a graph, then it is unique and spanned by the $2$-contractible edges. The third result of this paper is a construction of a graph which answers positively the question of Mohar whether regular infinite graphs with a unique Hamilton circle exist.

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Karl Heuer. 2017-01-21. Hamiltonicity in locally finite graphs: two extensions and a counterexample. https://arxiv.org/abs/1701.06029

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