Complexity of the Hamiltonian Cycle in Regular Graph Problem

Abstract The problem of deciding whether a 3-regular graph has a hamiltonian cycle (or path) was proved NP-complete. In this paper, we prove that for any fixed k ⩾ 3, deciding whether a k -regular graph has a hamiltonian cycle (or path) is a NP-complete problem. When the k -regular graph is planar, deciding whether the graph has a hamiltonian cycle (or path) was proved NP-complete for k = 3 and polynomial for k ⩾6. We prove that for k =4 and k =5 the problem is NP-complete.