Geodesic
Polyhedra with few 3-cuts are Hamiltonian
The electronic journal of combinatorics, Tome 26 (2019) no. 1
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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 2002 Jackson and 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 $k\ge 6$ 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.
DOI : 10.37236/7771
Classification : 05C45, 05C10, 05C40, 52B10 
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     author = {G. Brinkmann and C. T. Zamfirescu},
     title = {Polyhedra with few 3-cuts are {Hamiltonian}},
     journal = {The electronic journal of combinatorics},
     year = {2019},
     volume = {26},
     number = {1},
     doi = {10.37236/7771},
     zbl = {1409.05122},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/7771/}
}
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G. Brinkmann; C. T. Zamfirescu. Polyhedra with few 3-cuts are Hamiltonian. The electronic journal of combinatorics, Tome 26 (2019) no. 1. doi: 10.37236/7771

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