Colorings and girth of oriented planar graphs Original Research Article

Homomorphisms between graphs are studied as a generalization of colorings and of chromatic number. We investigate here homomorphisms from orientations of undirected planar graphs to graphs (not necessarily planar) containing as few digons as possible. We relate the existence of such homomorphisms to girth and it appears that these questions remain interesting even if we insist the girth of G is large, an assumption which makes the chromatic number easy to compute. In particular, we prove that every orientation of any large girth planar graph is 5-colorable and classify those digraphs on 3, 4 and 5 vertices which color all large girth oriented planar graphs.