Characterization of 1-Tough Graphs using Factors

For a graph $G$, let $odd(G)$ and $\omega(G)$ denote the number of odd components and the number of components of $G$, respectively. Then it is well-known that $G$ has a 1-factor if and only if $odd(G-S)\le |S|$ for all $S\subset V(G)$. Also it is clear that $odd(G-S) \le \omega(G-S)$. In this paper we characterize a 1-tough graph $G$, which satisfies $\omega(G-S) \le |S|$ for all $\emptyset \ne S \subset V(G)$, using an $H$-factor of a set-valued function $H:V(G) \to \{ \{1\}, \{0,2\} \}$. Moreover, we generalize this characterization to a graph that satisfies $\omega(G-S) \le f(S)$ for all $\emptyset \ne S \subset V(G)$, where $f:V(G) \to \{1,3,5, \ldots\}$.