## Hamilton cycles in plane triangulations.(English)Zbl 1012.05106

Summary: We extend Whitney’s theorem that every plane triangulation without separating triangles is Hamiltonian by allowing some separating triangles. More precisely, we define a decomposition of a plane triangulation $$G$$ into 4-connected ‘pieces,’ and show that if each piece shares a triangle with at most three other pieces then $$G$$ is Hamiltonian. We provide an example to show that our hypothesis that ‘each piece shares a triangle with at most three other pieces’ cannot be weakened to ‘four other pieces.’ As part of our proof, we also obtain new results on Tutte cycles through specified vertices in planar graphs.

### MSC:

 05C45 Eulerian and Hamiltonian graphs 05C38 Paths and cycles

### Keywords:

Hamilton cycle; plane triangulation; Tutte cycle
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### References:

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