Long cycles in graphs through fragments

Four basic Dirac-type sufficient conditions for a graph $G$ to be hamiltonian are known involving order $n$, minimum degree $\delta$, connectivity $\kappa$ and independence number $\alpha$ of $G$: (1) $\delta \geq n/2$ (Dirac); (2) $\kappa \geq 2$ and $\delta \geq (n+\kappa)/3$ (by the author); (3) $\kappa \geq 2$ and $\delta \geq \max\lbrace (n+2)/3,\alpha \rbrace$ (Nash-Williams); (4) $\kappa \geq 3$ and $\delta \geq \max\lbrace (n+2\kappa)/4,\alpha \rbrace$ (by the author). In this paper we prove the reverse version of (4) concerning the circumference $c$ of $G$ and completing the list of reverse versions of (1)-(4): (R1) if $\kappa \geq 2$, then $c\geq\min\lbrace n,2\delta\rbrace$ (Dirac); (R2) if $\kappa \geq 3$, then $c\geq\min\lbrace n,3\delta -\kappa\rbrace$ (by the author); (R3) if $\kappa\geq 3$ and $\delta\geq \alpha$, then $c\geq\min\lbrace n,3\delta-3\rbrace$ (Voss and Zuluaga); (R4) if $\kappa\geq 4$ and $\delta\geq \alpha$, then $c\geq\min\lbrace n,4\delta-2\kappa\rbrace$. To prove (R4), we present four more general results centered around a lower bound $c\geq 4\delta-2\kappa$ under four alternative conditions in terms of fragments. A subset $X$ of $V(G)$ is called a fragment of $G$ if $N(X)$ is a minimum cut-set and $V(G)-(X\cup N(X))\neq\emptyset$.