The authors show the following two results: {\parindent=5mm \begin{itemize}\item[1)]Let \(G\) be a graph of order \(n\geq 4k+3\) with \(\sigma_2 (G)\geq n\) and let \(F\) be a matching of size \(k\) in \(G\) such that \(G-F\) is 2-connected. Then \(G-F\) is hamiltonian or \(G\cong K_2 +(K_2\cup K_{n-4})\) or \(G\cong \bar{K_2} +(K_2\cup K_{n-4})\), where \(\sigma_2(G)\) denotes the minimum degree sum of two nonadjacent vertices. \item[2)]Let \(G\) be a graph of order \(n\geq 16k+1\) with \(\sigma_2(G)\geq n\) and let \(F\) be a set of \(k\) edges of \(G\) such that \(G-F\) is hamiltonian. Then \(G-F\) is either pancyclic or bipartite. \end{itemize}}