I'm new to the concept of graph minors, and they feel very much like homomorphisms to me. Initially I thought to myself, "why do graph theorists call a homomorphic graph a minor?" Of course, then I learned that there's a different concept of homomorphic graphs. So my question: is there nonetheless a relationship between these concepts? Does "$H$ is a minor of $G$" imply "$G$ is homomorphic into $H$?" I'm guessing the other direction is false; if so can somebody give a counterexample? Thank you!
Minors and homomorphisms are unrelated. Here are some examples to think about. There is a graph homomorphism from any bipartite graph to any non-discrete graph (any graph with an edge). Any cycle $C_n$ is a minor of any bigger cycle $C_m$ ($m\geq n$), but there is no homomorphism from a cycle of odd order to a cycle of even order.
In particular, $C_4$ is a minor of $C_5$, but there is no homomorphism from $C_5$ to $C_4$.