Hamiltonian paths required for Thomassen’s plus four modification

Determine whether every hypohamiltonian graph containing a cubic vertex automatically has Hamiltonian paths from the opposite boundary vertices after deleting the corresponding pairs of vertices, specifically whether the Hamiltonian paths from a to c in G - {b,d} and from b to d in G - {a,c} necessarily exist.

Background

The paper proves that Thomassen’s plus four modification preserves hypohamiltonicity under additional hypotheses. In particular, for a hypohamiltonian graph G with a cubic vertex in the local configuration H, the argument requires a Hamiltonian path from a to c in G - {b,d} and a Hamiltonian path from b to d in G - {a,c}.

The authors verify these paths for the specific graphs used to obtain their order-38 examples, but do not establish that their existence follows automatically from hypohamiltonicity. Whether the additional path conditions are redundant is therefore left unresolved.