an example for an arbitrary graph $G$ with even vertices which $\forall S \in V(G) , |N(S)|\geq |S| $ but there is no complete matching . I want to say an example for an arbitrary graph $G$ with even vertices which $\forall S \in V(G)  , |N(S)|\geq |S| $ but there is no complete matching .
I have tried so many shapes but I couldn't make right one!I try to make a connected one,if we drop this condition,we can consider a triangle with on isolated vertex.
it will be great if you give me some hint or guidance,thank you very much.
 A: Take three vertex-disjoint triangles. Add a tenth vertex, and edges joining it to one vertex in each of the three triangles.
It's easy to see that there is no perfect matching. The condition $|N(S)|\ge|S|$ is not quite so obvious, but it can be verified without examining $1023$ separate cases. (We could make the verification even easier by adding edges joining the tenth vertex to all the other vertices; it's still clear that there is no perfect matching.)
The vertex set is $V=V_1\cup V_2\cup V_3\cup\{u\}$ where $V_1,V_2,V_3$ are the vertex sets of the three triangles, and $u$ is the tenth vertex. Consider a subset $S\subseteq V$, and let $S_i=S\cap V_i$.
$|N(S_i)\cap V_i|\ge|S_i|$, with equality only if $|S_i|=0$ or $|S_i|=3$. Namely, if $|S_i|=1$ then $|N(S_i)\cap V_i|=2\gt|S_i|$, and if $|S_i|\ge2$ then $|N(S_i)\cap V_i|=3\ge|S_i|$.
$|N(S)\setminus\{u\}|\ge|N(S_1)\cap V_1|+|N(S_2)\cap V_2|+|N(S_3)\cap V_3|\ge|S_1|+|S_2|+|S_3|=|S\setminus\{u\}|$.
If $|S_i|=3$ for some $i$, then $u\in N(S)$ and $|N(S)|=|N(S)\setminus\{u\}|+1\ge|S\setminus\{u\}|+1\ge|S|$.
If $u\notin S$ then $|N(S)|\ge|N(S)\setminus\{u\}|\ge|S\setminus\{u\}|=|S|$.
If $S=\{u\}$ then $|N(S)|=3\gt|S|$.
If $u\in S$ and $S\ne\{u\}$ and $|S_i|\lt3$ for all $i$, then $|N(S)|\ge|N(S_1)\cap V_1|+|N(S_2)\cap V_2|+|N(S_3)\cap V_3|\gt|S_1|+|S_2|+|S_3|=|S\setminus\{u\}|$, so $|N(S)|\ge|S\setminus\{u\}|+1=|S|$.
