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

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