Counterexamples to a conjecture of Hoa on maximal non-Hamiltonian graphs

A graph G is said to be maximal non-Hamiltonian if G is non-Hamiltonian, but $G+e$ is Hamiltonian for every nonedge $e$ of $G\mathrm{.}$ In 1994, Vu Dinh Hoa conjectured that if $C$ is a longest cycle of a maximal non-Hamiltonian graph $G,$ then $G-V(C)$ is a complete graph. We disprove this conjecture by constructing a counterexample of every order $n\ge 56.$ We also pose several related open problems.