Removable matchings and hamiltonian cycles

For a graph GG, let σ2(G)σ2(G) denote the minimum degree sum of two nonadjacent vertices (when GG is complete, we let σ2(G)=∞σ2(G)=∞). In this paper, we show the following two results: (i) Let GG be a graph of order n≥4k+3n≥4k+3 with σ2(G)≥nσ2(G)≥n and let FF be a matching of size kk in GG such that G−FG−F is 2-connected. Then G−FG−F is hamiltonian or G≅K2+(K2∪Kn−4)G≅K2+(K2∪Kn−4) or G≅K2¯+(K2∪Kn−4); (ii) Let GG be a graph of order n≥16k+1n≥16k+1 with σ2(G)≥nσ2(G)≥n and let FF be a set of kk edges of GG such that G−FG−F is hamiltonian. Then G−FG−F is either pancyclic or bipartite. Examples show that first result is the best possible.