A note on the almost $1$-tough graph

Abstract

A graph $G$ is almost $1$-tough if $c(G-S)\leq |S|+1$ for any$S\subseteq V(G)$ , where $c(G-S)$ is the number of components of$G-S$. Let $F: V(G)\rightarrow \{\{1\},\{0,2\}\}$ be a set-valuedfunction and $F^{-1}(1):=\{v\in V(G): f(v)=\{1\}\}$. A spanningsubgraph $H$ of $G$ is called an $F$-factor if $d_{H}(v)\in F(v)$for all $v\in V(G)$. It is interesting to know whether a graph isalmost $1$-tough or a graph has an $F$-factor.  In this note, weestablish a lower bound on the size (resp. the spectral radius) toensure a graph to be almost $1$-tough. This also provide asufficient condition for the existence of an $F$-factor for whichevery $F: V(G)\rightarrow \{\{1\},\{0,2\}\}$ with $|F^{-1}(1)|$ evenin a graph.