# Endomorphism of a graph and a maximal clique

Let $$G= (V, E)$$ be a simple graph satisfy the following:

• Independence number is $$3$$,
• Clique number is $$\frac{n^2}{4}$$,
• Number of vertices is $$n^2$$, where $$n$$ is even.

Let $$f \in$$ End$$(G)$$, where End$$(G)$$ is the collection of all homomorphism from $$G$$ to $$G$$.

I want to show that atmost three clique of maximum size can be mapped under $$f$$ to a single clique of maximum size.

Since independence number of a graph is $$3$$ so that atmost three vertices can be mapped into a single single vertex. Also, it is easy to verified that image of maximal clique is a maximal clique under a homomorphism. I am stuck here, how to prove that atmost three clique of maximum size can be mapped to a single clique of maximum size.