Oriented Chromatic Number of Cartesian Products Pm □ Pn and Cm □ Pn

Abstract We consider oriented chromatic number of Cartesian products of two paths Pm □ Pn and of Cartesian products of paths and cycles, Cm □ Pn. We say that the oriented graph G→ \vec G is colored by an oriented graph H→ \vec H if there is a homomorphism from G→ \vec G to H→ \vec H . In this paper we show that there exists an oriented tournament H→10 {\vec H_{10}} with ten vertices which colors every orientation of P8 □ Pn and every orientation of Cm □ Pn, for m = 3, 4, 5, 6, 7 and n ≥ 1. We also show that there exists an oriented graph T→16 {\vec T_{16}} with sixteen vertices which colors every orientation of Cm □ Pn.