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  implies , where is a property of graphs that is NP-hard and is a cycle structure property of graphs that is also NP-hard. An example of such a theorem is the well-known ChvtalErd?s Theorem, which states that every graph G with ??? is hamiltonian. Here ? is the vertex connectivity of G and ? is the cardinality of a largest set of independent vertices of G. In another paper Chvtal 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 ? 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 a well-known theorem of Jung (Ann. Discrete Math. 3 (1978) 129) for graphs on 16 or more vertices.
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