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Non-Hamiltonian 1-tough triangulations with disjoint separating triangles. (English) Zbl 1443.05108

Summary: In this note, we consider triangulations of the plane. K. Ozeki and C. T. Zamfirescu [Discrete Math. 341, No. 7, 1900–1902 (2018; Zbl 1387.05141)] asked whether there are non-Hamiltonian 1-tough triangulations in which every two separating triangles are disjoint. We answer this question in the affirmative and strengthen a result of T. Nishizeki [ibid. 30, 305–307 (1980; Zbl 0442.05046)] by proving that there are infinitely many non-Hamiltonian 1-tough triangulations with pairwise disjoint separating triangles.

MSC:

05C45 Eulerian and Hamiltonian graphs
05C40 Connectivity
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