Connectivity and Hamiltonian connectedness of graphs (Q1801761)

Let \(\sigma_ m(G)=\min\left\{\sum^ m_{i=1}d(v_ i)\mid\{v_ 1,v_ 2,\ldots,v_ m\}\text{ is an independent set}\right\}\). Theorem: Let \(G\) be a 2-connected graph of order \(n\) with connectivity \(\kappa\), and let \(\sigma_ 3\geq n+\kappa+1\). If \(u,v\in V(G)\) and \(\{u,v\}\) is not a vertex cut of \(G\), then \(G\) contains a Hamiltonian \(u\)-\(v\) path.