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In mathematics, Tait's conjecture states that " Every 3-connected planar cubic graph has a Hamiltonian cycle ( along the edges ) through all its vertices ".

It was proposed by and disproved by, who constructed a counterexample with 25 faces, 69 edges and 46 vertices.

Several smaller counterexamples, with 21 faces, 57 edges and 38 vertices, were later proved minimal by.