# Critical graphs and endomorphisms

We call a finite simple undirected graph $G=(V,E)$ critical if $\chi(G\setminus\{v\}) < \chi(G)$ for all $v\in V$.

If $G$ is critical then for any graph homomorphism $f:G\to G$ we have that $f$ is surjective, and therefore bijective. (The reason is that whenever $f: G\to H$ is a graph homomorphism, then $\chi(G)\leq \chi(H)$.)

What is an example of a graph $G$ that is not critical, but every graph homomorphism $f:G\to G$ is surjective?

Take any two nonisomorphic $k$-strongly critical graphs $G$ and $H$ with the same number of vertices (e.g., the Mycielski construction on a triangle and the Hajos construction on two copies of $K_4$, for $k=4$). Their disjoint union fits the goal, since there are no homomorphisms from $G$ to $H$ and vice versa.