Examples
The oriented chromatic number of a directed 5-cycle is five. If the cycle is colored by four or fewer colors, then either two adjacent vertices have the same color, or two vertices two steps apart have the same color. In the latter case, the edges connecting these two vertices to the vertex between them are inconsistently oriented: both have the same pair of colors but with opposite orientations. Thus, no coloring with four or fewer colors is possible. However, giving each vertex its own unique color leads to a valid oriented coloring.
Read more about this topic: Oriented Coloring
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