Homomorphisms of triangle-free graphs without a -minor

In the course of extending Grötzsch’s Theorem, we prove that every triangle-free graph without a K 5 -minor is 3-colorable. It has been recently proved that every triangle-free planar graph admits a homomorphism to the Clebsch graph. We also extend this result to the class of triangle-free graphs without a K 5 -minor. This is related to some conjectures which generalize the Four-Color Theorem. While we show that our results cannot be extended directly, we conjecture that every K 6 -minor-free graph of girth at least 5 is 3-colorable.