On empty Hom-sets between graphs

In many categories such as Set, Group, Top (topological spaces) there is a morphism between any two objects, usually in both directions. (If one object is “empty”, like the empty space, or the empty set, there is the “empty” morphism from \emptyset to the other object, but not the other way round.)

However in the category Graph it is possible that non-empty graphs G,H have no graph homomorphism between them in either direction:

Let K_3 be the complete graph on 3 points and let H be any triangle-free graph with \chi(H) > 3, for instance the Grötzsch graph, which has chromatic number 4.

Since H is triangle-free, there is no graph homomorphism \varphi: K_3 \to H and if there were a homomorphism \eta : H \to K_3, this map would be an n-coloring of H for some n \leq 3.

About dominiczypen

I'm interested in general topology, order theory, and graph theory. This link takes you to my preprints on arXiv.
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