On universal graphs for planar oriented graphs of a given girth Original Research Article

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 K has a homomorphism to the same H′, then H′ is K -universal. Let Pk denote the class of orientations of planar graphs with girth at least k. Clearly, P3 ⊃ P4 ⊃ P5… We discuss the existence of Pk-universal graphs with special properties. It is known (see Raspaud and Sopena, 1994) that there exists a P3-universal graph on 80 vertices. We prove here that 1. (1) there exist no planar P4-universal graphs; 2. (2) there exists a planar P16-universal graph on 6 vertices; 3. (3) for any k, there exist no planar Pk-universal graphs of girth at least 6; 4. (4) for any k, there exists a P40k-universal graph of girth at least k + 1