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Two new sufficient conditions for generalized cycles (including Hamilton anddominating cycles as special cases) in an arbitrary k-connected graph(k=1,2,...) are derived, which prove the truth of Bondy's (1980) famousconjecture for some variants significantly improving the result expected by thegiven hypothesis. Similarly, two new lower bounds for the circumference (thelength of a longest cycle) are established for the reverse hypothesis proposedby Jung (2001).

A Note on Large Cycles in Graphs Around Conjectures of Bondy and Jung