Antisymmetric flows and strong colourings of oriented graphs

The homomorphisms of oriented or undirected graphs, the oriented chromatic number, the relationship between acyclic colouring number and oriented chro matic number, have been recently intensely studied. For the purpose of dual ity, we define the notions of strong-oriented colouring and antisymmetric-f low. An antisymmetric- Row is a flow with values in an additive abelian gro up which uses no opposite elements of the group. We prove that the strong-o riented chromatic number <(chi)over right arrow>(s) (as the modular version of oriented chromatic number) is bounded for planar graphs.