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) $\delta \ge n\mathrm{/}2$ (Dirac); (2) $\kappa \ge 2$ and $\delta \ge (n+\kappa )\mathrm{/}3$ (by the author); (3) $\kappa \geq 2$and$\delta \geq \max\lbrace (n+2)/3,\alpha \rbrace$ (Nash-Williams); (4) $\kappa \ge 3$ and $\delta \ge \max \{(n+2\kappa )\mathrm{/}4,\alpha \}$ (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 $\kappa \ge 2$, then $c\ge \min \{n,2\delta \}$ (Dirac); (R2) if $\kappa \ge 3$, then $c\geq\min\lbrace n,3\delta -\kappa\rbrace$(bytheauthor);(R3)if$\kappa\geq 3$and$\delta\geq \alpha$,then$c\geq\min\lbrace n,3\delta-3\rbrace$(VossandZuluaga);(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 $c\ge 4\delta -2\kappa$ 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 $V(G)-(X\cup N(X))\mathrm{\ne }\mathrm{\varnothing }$.