Dirac-type generalizations concerning large cycles in graphs

Bondy conjectured a common generalization of various results in hamiltonian graph theory concerning Hamilton and dominating cycles by introducing a notion of P D λ -cycles (cycles that dominate all paths of lengths at least λ ). We show that the minimum degree version of Bondy’s conjecture is true (along with the reverse version) if P D λ -cycles are replaced by C D λ -cycles (cycles that dominate all cycles of lengths at least λ ). Fraisse proved a minimum degree generalization including a theorem of Nash-Williams for Hamilton cycles as a special case. We present the reverse version of this result including a theorem of Voss and Zuluaga as a special case. Two earlier less known results (due to the author) are crucial for the proofs of these results. All results are sharp in all respects. A number of possible similar generalizations are conjectured as well.