Polynomial algorithms that prove an NP-Hard hypothesis implies an NP-hard conclusion

A number of results in hamiltonian graph theory are of the form P1 implies P2, where P1 is a property of graphs that is NP-hard and P2 is a cycle structure property of graphs that is also NP-hard. Such a theorem is the well-known Chvátal-Erdös Theorem, which states that every graph G with α ≤ κ is hamiltonian. Here κ is the vertex connectivity of G and α is the cardinality of a largest set of independent vertices of G. In another paper Chvátal points out that the proof of this result is in fact a polynomial time construction that either produces a Hamilton cycle or a set of more than κ independent vertices. In this note we point out that other theorems in hamiltonian graph theory have a similar character. In particular, we present a constructive proof of the well-known theorem of Jung [12] for graphs on 16 or more vertices.