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

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. An example of such a theorem is the well-known Chvátal–Erdős 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 a well-known theorem of Jung (Ann. Discrete Math. 3 (1978) 129) for graphs on 16 or more vertices.