Colored Homomorphisms of Colored Mixed Graphs

The homomorphisms of oriented or undirected graphs, the oriented chromatic number, the relationship between acyclic coloring number and oriented chromatic number, have been recently studied. Improving and combining earlier techniques of N. Alon and T. H. Marshall (1998, J. Algebraic Combin.8, 5–13) and A. Raspaud and E. Sopena (1994, Inform. Process. Lett.51, 171–174) we prove here a general result about homomorphisms of colored mixed graphs which implies all these earlier results about planar graphs. We also determine the exact chromatic number of colored mixed trees. For this, we introduce the notion of colored homomorphism for mixed graphs containing both colored arcs and colored edges.