On universal graphs for planar oriented graphs of a given girth

The oriented chromatic number o(H) of an oriented graph H is defined to be the minimum order of an oriented graph H ' such that H has a homomorphism to H' . If each graph in a class ~ has a homomorphism to the same H' , then H ' is ~-universal. Let ~k denote the class of orientations of planar graphs with girth at least k. Clearly, ~3 ~ ~4 ~ ~5... We discuss the existence of ~k-universal graphs with special properties. It is known (see Raspaud and Sopena, 1994) that there exists a ~3-universal graph on 80 vertices. We prove here that (1) there exist no planar ~4-universal graphs; (2) there exists a planar ~16-universal graph on 6 vertices; (3) for any k, there exist no planar ~k-universal graphs of girth at least 6; (4) for any k, there exists a ~40k-universal graph of girth at least k + 1. (~ 1998 Elsevier Science B.V. All rights reserved