Hamiltonicity in locally finite graphs: two extensions and a counterexample

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.

arXiv comment26 pages, 6 figures; minor revision of Section 3, especially by adding Claim 3.4. Correction of a small error in the proof of Theorem 1.8 by including the new Lemma 4.4. A remark about a follow-up paper to this article is added

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