Abstract
A graph is hamiltonian if it has a hamiltonian cycle. It is well-known that Tutte proved that any 4-connected planar graph is hamiltonian. It is also well-known that the problem of determining whether a 3-connected planar graph is hamiltonian is NP-complete. In particular, Chvátal and Wigderson had independently shown that the problem of determining whether a maximal planar graph is hamiltonian is NP-complete. A classical theorem of Whitney says that any maximal planar graph with no separating triangles is hamiltonian, where a separating triangle is a triangle whose removal separates the graph. Note that if a planar graph has separating triangles, then it can not be 4-connected and therefore Tutte's result can not be applied. In this paper, we shall prove that any maximal planar graph with only one separating triangle is still hamiltonian.
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Chen, C. Any Maximal Planar Graph with Only One Separating Triangle is Hamiltonian. Journal of Combinatorial Optimization 7, 79–86 (2003). https://doi.org/10.1023/A:1021998507140
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DOI: https://doi.org/10.1023/A:1021998507140