On Pure Degree Sequence Manipulations Forcing Long Cycles in Graphs

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.