Given two graphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$, an homomorphism of $G_1$ to $G_2$ is a function $f:V_1 \rightarrow V_2$ such $(v,w) \in E_1 \rightarrow (f(v),f(w)) \in E_2$. We establish that $G_1$ is homomorph to $G_2$ if and only if there's an homomorphism from $G_1$ to $G_2$.
Be $C_n$ a graph such is a simple loop of $n \geq 3$ vertex. Be $p$ and $q$ integers such $p,q \geq 3$
Proof that if $p$ and $q$ are odd, then $C_p$ is homomorph to $C_q$ if and only if $p \geq q$
I think I know how to proof that if $p$ is greater than $q$ then the homomorphism exists. But I find trouble proofing that if $p < q$ then there are not $any$ homomorphisms such the condition met.
Write the cycle of $C_p$ as $(v_1,\ldots,v_p)$ (so $v_i$ is adjacent to $v_{i+1}$ and $v_1$ adjacent to $v_p$).
If $f:C_p\to C_q$ is a graph homomorphism, then notice that $(f(v_1),f(v_2),\ldots,f(v_p))$ corresponds to a closed walk of length $p$ in $C_q$.
So if $p<q$, can this walk use all edges of $C_q$? And if it doesn't, can $p$ be odd?