Motivated by a conjecture of Grünbaum and problem Katona, Kostochka, Pach, Stechkin, both dealing with non-Hamiltonian $n$-vertex graphs their $(n-2)$-cycles, we investigate $K_2$-Hamiltonian graphs, i.e., in which the removal any pair adjacent vertices yields Hamiltonian graph. In this first part, prove structural properties show that there exist infinitely many cubic 3-edge-colorable non-3-ed...