In graph theory, oriented graph coloring is a special type of graph coloring. Namely, it is
an assignment of colors to vertices of an oriented graph that
is proper: no two adjacent vertices get the same color, and
is consistently oriented: if vertices and have the same color, and vertices and have the same color, then and cannot both be edges in the graph.
Equivalently, an oriented graph coloring of a graph G is an oriented graph H (whose vertices represent colors and whose arcs represent valid orientations between colors) such that there exists a homomorphism from G to H.
An oriented chromatic number of a graph G is the fewest colors needed in an oriented coloring;
it is usually denoted by . The same definition can be extended to undirected graphs, as well, by defining the oriented chromatic number of an undirected graph to be the largest oriented chromatic number of any of its orientations.[1]
^Kostochka, A. V.; Sopena, E.; Zhu, X. (1997), "Acyclic and oriented chromatic numbers of graphs" (PDF), Journal of Graph Theory, 24 (4): 331–340, doi:10.1002/(SICI)1097-0118(199704)24:4<331::AID-JGT5>3.0.CO;2-P, MR 1437294.
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