homomorphism between Petersen graph and $C_5$

Question

Why $\text{Hom}(\text{Petersen graph}, C_5)=\emptyset $ ?

Answer 1

Assuming that by $\text{Hom}$ you mean the set of graph homomorphisms of graphs, and that the graphs in question do not have loops.

First of all, consider homomorphisms from $C_5$ to $C_5$. It should be easy to observe that all such homomorphisms are necessarily bijective.

Assume that $\text{Hom}(\text{Petersen graph},C_5)$ set is nonempty, and there is a homomorphism $h$ in that set. Choose the 5-cycle $ABCDE$ in the Petersen graph. Let's set $h(A) =: V, h(B) =: W, h(C) =: X, h(D) =: Y, h(E) =: Z$. Note that $VWXYZ$ are the vertices of $C_5$ in order, because of our previous observation.

Now the Petersen graph contains another 5-cycle $ABCFG$, where $F, G$ are not the same as $D, E$. By necessity, $h(F) = Y, h(G) = Z$.

But now the Petersen graph contains another 5-cycle $DEAGH$, where $H$ is a new vertex. Since $H$ is adjacent to $D$ and $G$, $h(H)$ must be adjacent to $h(D) = Y$ and $h(G) = Z$. But there is no such vertex in $C_5$ adjacent to $Y$ and $Z$ simultaneously. This is a contradiction, so our assumption that $h$ exists is false.