Tait's conjecture states that "Every
polyhedron has a
Hamiltonian cycle (along the edges) through all its
vertices". It was proposed in 1886 by
P. G. Tait[?] and disproved in 1946, when
W. T. Tutte constructed a counterexample with 25 faces, 69 edges and 46 vertices.
The conjecture could have been significant, because if true, it would have implied the four color theorem.