On the chromatic number of a random 5-regular graph

It was only recently shown by Shi and Wormald, using the differential equation method to analyse an appropriate algorithm, that a random 5-regular graph asymptotically almost surely has chromatic number at most 4. Here, we show that the chromatic number of a random 5-regular graph is asymptotically almost surely equal to 3, provided a certain four-variable function has a unique maximum at a given point in a bounded domain. We also describe extensive numerical evidence which strongly suggests that the latter condition holds. The proof applies the small subgraph conditioning method to the number of locally rainbow balanced 3-colourings, where a colouring is balanced if the number of vertices of each colour is equal, and locally rainbow if every vertex is adjacent to at least one vertex of each of the other colours.