Toughness, degrees and 2-factors

In this paper we generalize a Theorem of Jung which shows that 1-tough graphs with View the MathML sourceδ(G)⩾|V(G)|-42 are hamiltonian. Our generalization shows that these graphs contain a wide variety of 2-factors. In fact, these graphs contain not only 2-factors having just one cycle (the hamiltonian case) but 2-factors with kk cycles, for any kk such that View the MathML source1⩽k⩽n-164.