$K_2$-Hamiltonian Graphs: I

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-edge-colorable variety. fact, chromatic index 4 (such as Petersen's graph) are subset critical snarks. On other hand, it is proven maximum degree exist. Several operations conserving $K_2$-Hamiltonicity described, one leads to result exists an infinite family which, asymptotically, quarter has property removing such vertex We extend celebrated Tutte showing every planar graph minimum at least Hamiltonian. Finally, $K_2$-traceable discuss open problems.