Let $\Gamma$ be any simple undirected graph whose independence number is $3$ and the clique number is $r$. Suppose $f$ be an endomorphism on $\Gamma$ such that $f(x) = x$ for all $x \in {\rm im} f$. I know that the image of any clique of size $r$ is a clique of size $r$. I am unable to prove this property:
I guess that at most three cliques of size $r$ can map into a clique of size $r$.
I do not know this is true or not. Thanks in advance for your kind help.
It is false.
Consider the cycle graph $\Gamma = C_6$ and number the vertices such that 1 ~ 2 ~ 3 ~ 4 ~ 5 ~ 6 ~ 1. Then $r = 2$ and the independence number of $\Gamma$ is $3$. Let $f$ map odd numbered vertices to $1$ and even numbered vertices to $2$. Then $f$ satisfies the given constraints, but all $6$ size-$2$ cliques of the graph map to the clique $\{1,2\}$.