Homomorphically equivalent but not a subgraph of each other

Question

Referring to the definition of homomorphic equivalence here, is it possible that two homomorphically equivalent graphs are not a subgraph of each other?

Answer 1

The cycles $C_4$ and $C_6$ are such graphs. As they are not subgraphs of each other, we only have to show that there are homomorphisms $C_4\to C_6$ and $C_6\to C_4.$

In fact, whenever $n\ge 2$ and $m\ge 3,$ we have an homomorphism $C_{2n} \to C_m.$ Let's denote the vertices of $C_k$ by $v_1,v_2,\dots,v_k$ in the natural order ($v_1$ has neighbors $v_2$ and $v_k,$ etc.). We map a vertex $v_i$ of $C_{2n}$ to $v_1\in V(C_m)$ if $i$ is odd, and to $v_2\in V(C_m)$ otherwise.