The 2‐hamiltonian cubes of graphs

This paper deals with the problem of characterizing the pairs of vertices x,y in a connected graph G such that G 3 ‐ { x,y } is hamiltonian, where G 3 is the cube of G. It is known that the cube G 3 is 2‐hamiltonian if G is 2‐connected. In this paper, we first prove the stronger result that G 3 ‐ { x,y } is hamiltonian if either x or y is not a cut‐vertex of G , and then proceed to characterize those cut‐vertices x and y of G such that G 3 ‐{ x,y } is hamiltonian. As a simple consequence of these, we obtain Schaar's characterization of a connected graph G such that G 3 is 2‐hamiltonian.