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Let $k \geq 2$ be an integer. We say that a graph $G$ is $(K_2 \cupkK_1)$-free if it does not contain $K_2 \cup kK_1$ as an induced subgraph.Recently, Shi and Shan conjectured that every $1$-tough and $2k$-connected$(K_2 \cup kK_1)$-free graph is hamiltonian. In this paper, we solve thisconjecture by proving the statement; every $1$-tough and $k$-connected $(K_2\cup kK_1)$-free graph with minimum degree at least $\frac{3(k-1)}{2}$ ishamiltonian or the Petersen graph.

Some conditions for hamiltonian cycles in 1-tough $(K_2 \cup kK_1)$-free graphs