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According to Wikipedia and other sources, Tait's conjecture would have had significant effect on solving the FCP had it been true.

Are there other mathematical connections between FCP and hamiltonicity?

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up vote 1 down vote accepted

MR0975998 (90d:05156) Fleischner, H. The prism of a 2-connected, planar, cubic graph is Hamiltonian (a proof independent of the four colour theorem). Graph theory in memory of G. A. Dirac (Sandbjerg, 1985), 141–170, Ann. Discrete Math., 41, North-Holland, Amsterdam, 1989.

The review says, in part,

Using the 4CT, the result of the title was shown by M. Rosenfeld and D. W. Barnette [Discrete Math. 5 (1973), 389–394; MR0321806 (48 #173)]. Now the author offers a much longer and more intricate proof, which, however, is 4CT-free.

[So, the connection here is that the first proof of Hamiltonicity relied on the Four Color Theorem, but the newer proof didn't.]

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