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Note that this problem is affirmative for $k \\in \\{1,2,3\\}$ by the known results. In this paper, we show that for each integer $k \\geq 4$, if $G$ is a $1$-tough and $(k-1)$-connected $(P_2 \\cup kP_1)$-free graph with $|V(G)| \\ge k^2+k+1$ and $\\delta(G) \\ge k$, then $G$ is hamiltonian. 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