Properties
An oriented coloring can exist only for a directed graph with no loops or directed 2-cycles. For, a loop cannot have different colors at its endpoints, and a 2-cycle cannot have both of its edges consistently oriented between the same two colors. If these conditions are satisfied, then there always exists an oriented coloring, for instance the coloring that assigns a different color to each vertex.
If an oriented coloring is complete, in the sense that no two colors can be merged to produce a coloring with fewer colors, then it corresponds uniquely to a graph homomorphism into a tournament. The tournament has one vertex for each color in the coloring. For each pair of colors, there is an edge in the colored graph with those two colors at its endpoints, which lends its orientation to the edge in the tournament between the vertices corresponding to the two colors. Incomplete colorings may also be represented by homomorphisms into tournaments but in this case the correspondence between colorings and homomorphisms is not one-to-one.
Undirected graphs of bounded genus, bounded degree, or bounded acyclic chromatic number also have bounded oriented chromatic number.
Read more about this topic: Oriented Coloring
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