A note on the almost 1-tough graph

A graph G is almost 1-tough if c(G-S) ? |S| + 1 for any S ? V(G), where c(G-S) is the number of components of G-S. Let F: V(G) ? {{1}, {0, 2}} be a set-valued function and F⁻¹(1) := {v ? V(G): ? (v) = {1}} . A spanning subgraph H of G is called an F-factor if d_H (v) ? F(v) for all v ? V(G). It is interesting to know whether a graph is almost 1-tough or a graph has an F-factor. In this note, we establish a lower bound on the size (resp. the spectral radius) to ensure a graph to be almost 1-tough. This also provide a sufficient condition for the existence of an F-factor for which every F: V(G) ? {{1}, {0, 2}} with | F⁻¹ (1)| even in a graph.