High-Girth Graphs Avoiding a Minor are Nearly Bipartite

Let H be a fixed graph. We show that any H-minor free graph G of high enough girth has circular chromatic number arbitrarily close to two. Equivalently, each such graph G admits a homomorphism to a large odd circuit. In particular, graphs of high girth and of bounded genus, or of bounded tree width, are “nearly bipartite” in this sense. For example, any planar graph of girth at least 16 admits a homomorphism to a pentagon. We also obtain tight bounds on the girth of G in a few specific cases of small forbidden minors H.