Finding a Lower Bound for k-Unbounded Hamiltonian Cycles

Abstract: Methods to determine the existence of Hamiltonian Cycles in graphs have been extensively studied. However, little research has been done following cases when no Hamiltonian Cycle exists. Let a vertex be "unbounded" if it is visited more than once in a path. Furthermore, let a k-Unbounded Hamiltonian Cycle be a path with finite length that visits every vertex, has adjacent start and end vertices, and contains k unbounded vertices. We consider a novel variant of the Hamiltonian Cycle Problem in which the objective is to find an m-Unbounded Hamiltonian Cycle where m is the minimum value of k such that a k-Unbounded Hamiltonian Cycle exists. We first consider the task on well-known non-Hamiltonian graphs. We then provide an exponential-time brute-force algorithm for the determination of an m-Unbounded Hamiltonian Cycle and discuss approaches to solve the variant through transformations to the Hamiltonian Cycle Problem and the Asymmetric Traveling Salesman Problem. Finally, we present a polynomial-time heuristic for the determination of an m-Unbounded Hamiltonian Cycle that is also shown to be an effective heuristic for the original Hamiltonian Cycle Problem.