On Hamiltonian cycles and Hamiltonian paths

A Hamiltonian cycle is a spanning cycle in a graph, i.e., a cycle through every vertex, and a Hamiltonian path is a spanning path. In this paper we present two theorems stating sufficient conditions for a graph to possess Hamiltonian cycles and Hamiltonian paths. The significance of the theorems is discussed, and it is shown that the famous Ore's theorem directly follows from our result.