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Polyhedra with few 3-cuts are Hamiltonian. (English) Zbl 1409.05122
Summary: In 1956, Tutte showed that every planar 4-connected graph is Hamiltonian. In this article, we will generalize this result and prove that polyhedra with at most three \(3\)-cuts are Hamiltonian. In [J. Graph Theory 41, No. 2, 138–150 (2002; Zbl 1012.05106)], B. Jackson and X. Yu have shown this result for the subclass of triangulations. We also prove that polyhedra with at most four \(3\)-cuts have a Hamiltonian path. It is well known that for each \(k\geq 6\) non-Hamiltonian polyhedra with \(k\) \(3\)-cuts exist. We give computational results on lower bounds on the order of a possible non-Hamiltonian polyhedron for the remaining open cases of polyhedra with four or five \(3\)-cuts.

MSC:
05C45 Eulerian and Hamiltonian graphs
05C10 Planar graphs; geometric and topological aspects of graph theory
05C40 Connectivity
52B10 Three-dimensional polytopes
Software:
plantri
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References:
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