### Abstract

When considering proper colourings of oriented graphs, one possible approach is to insist that the colouring be consistent with the orientation: all edges between two colour classes should be oriented in the same way. The oriented chromatic number of an oriented graph is the least $k$ so that there is a homomorphism from the oriented graph to some tournament of order $k$. In this talk, I will present new work with Nir on the oriented chromatic number of random models of oriented graphs of bounded degree. Previous extremal results can be used to bound the oriented chromatic number of a random $d$-regular oriented graph between $\Omega(\sqrt{2}^d)$ and $O(d^2 2^d)$. Using colourings by doubly-regular tournaments, we improve the upper bound to $O(\sqrt{e}^d)$. I will present these results and discuss an optimization result for functions over doubly stochastic matrices that extends the optimization result of Achlioptas and Naor that was central to results on chromatic numbers of unoriented random graphs.