Polyhedra with few 3-cuts are Hamiltonian.

*(English)*Zbl 1409.05122Summary: 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 |

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\textit{G. Brinkmann} and \textit{C. T. Zamfirescu}, Electron. J. Comb. 26, No. 1, Research Paper P1.39, 16 p. (2019; Zbl 1409.05122)

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##### References:

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