  A number of new sufficient conditions for generalized cycles (large cycles
including Hamilton and dominating cycles as special cases) in an arbitrary
$k$-connected graph $(k=1,2,...)$ and new lower bounds for the circumference
(the length of a longest cycle) are derived, inspiring a number of
modifications of famous conjectures of Bondy (1980) and Jung (2001). All
results (both old and new) are shown to be best possible in a sense based on
three types of sharpness indicating the intervals in which the result is sharp
and the intervals in which the result can be further improved. In addition, the
presented modifications cannot be derived directly from Bondy's and Jung's
conjectures as special cases.
