Counting Hamiltonian cycles in planar triangulations

Hakimi, Schmeichel, and Thomassen [J. Graph Theory, 1979 ] conjectured that every 4-connected planar triangulationG on n vertices has at least 2(n−2)(n−4) Hamiltonian cycles, with equality if and only if G is a double wheel. In this paper, we show that every 4-connected planar triangulation on n vertices has Ω(n) Hamiltonian cycles. Moreover, we show that if G is a 4-connected planar triangulation on n vertices and the distance between any two vertices of degree 4 in G is at least 3, then G has 2 )

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