On Generalizing a Theorem of Jung

For a graph \(G\), let \(\sigma_k = \min \{\sum_{i=1}^{k} d(v_i) \mid \{v_1, \ldots, v_k\}\) { is an independent set
of vertices in } G\}. Jung proved that every \(1\)-tough graph \(G\) with \(|V(G)| = n \geq 11\) and \(\sigma_2 > n-4\) is hamiltonian. This result is generalized as follows: if \(G\) is a \(1\)-tough graph with \(|V(G)| = n \geq 3\) such that \(\sigma_3 > n\) and for all \(x, y \in V(G)\), \(d(x,y) = 2\) implies \(\max\{d(x), d(y)\} \geq \frac{1}{2}(n-4)\), then \(G\) is hamiltonian. It is also shown that the condition \(\sigma_3 \geq n\), in the latter result, can be dropped if \(G\) is required to be \(3\)-connected and to have at least \(35\) vertices.