  The well-known Hamiltonian sufficient conditions, proposed by Dirac, Faudree
et al., P\'osa, Bondy, Chv\'atal are based on pure degree manipulations without
any additional conditions. In this paper, we present two new types of pure
degree manipulations that produce relatively simpler (in view of verification)
results. The reverse versions (long-cycle versions) of the obtained results are
presented as well. The importance of pure degree manipulations is motivated by
their amenability (as starting points) to generalizations and new ideas in a
great variety of ways.
