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A number of new sufficient conditions for generalized cycles (large cyclesincluding 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 ofmodifications of famous conjectures of Bondy (1980) and Jung (2001). Allresults (both old and new) are shown to be best possible in a sense based onthree types of sharpness indicating the intervals in which the result is sharpand the intervals in which the result can be further improved. In addition, thepresented modifications cannot be derived directly from Bondy's and Jung'sconjectures as special cases.

Large Cycles in Graphs Around Conjectures of Bondy and Jung -- Modifications and Sharpness