A bipartite graph $G$ such that for any partition $X$ and $Y$, $\alpha(G)> \max \{X,Y\}$

Question

The question is

Construct a bipartite graph $G$ such that for any partition $X$ and $Y$, $\alpha(G)> \max \{X,Y\}$. Explain.

I did this graph:

Here it works, $\alpha (G) =9$ (marked in blue box), and each partition is $7$. However, if we flip the red circled vertices so that the $3$ top vertices become in $Y$-partition $Y$ will be equal to $9$, which violates what I want to prove. Is there any way to fix my graph?

Answer 1

When you flip the graph then y will have cardinality 9 and maximal independant set will become 9 . So it proves what you are trying to prove.