On Longest Cycle $C$ of a graph $G$ via Structures of $G-C$

Two sharp lower bounds for the length of a longest cycle $C$ of a graph $G$ are presented in terms of the lengths of a longest path and a longest cycle of $G-C$, denoted by $\overline{p}$ and $\overline{c}$, respectively, combined with minimum degree $\delta$: (1) $|C|\geq(\overline{p}+2)(\delta-\overline{p})$ and (2) $|C|\geq(\overline{c}+1)(\delta-\overline{c}+1)$.