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In [Graphs Combin.~24 (2008) 469--483.], the third author and the fifthauthor conjectured that if $G$ is a $k$-connected graph such that$\sigma_{k+1}(G) \ge |V(G)|+\kappa(G)+(k-2)(\alpha(G)-1)$, then $G$ contains aHamiltonian cycle, where $\sigma_{k+1}(G)$, $\kappa(G)$ and $\alpha(G)$ are theminimum degree sum of $k+1$ independent vertices, the connectivity and theindependence number of $G$, respectively. In this paper, we settle thisconjecture. This is an improvement of the result obtained by Li: If $G$ is a$k$-connected graph such that $\sigma_{k+1}(G) \ge |V(G)|+(k-1)(\alpha(G)-1)$,then $G$ is Hamiltonian. The degree sum condition is best possible.

A degree sum condition on the order, the connectivity and the independence number for Hamiltonicity