  A graph is k-total colourable if there is an assignment of k different
colours to the vertices and edges of the graph such that no two adjacent nor
incident elements receive the same colour. The total chromatic number of some
direct product graphs are determined. In particular, a sufficient condition is
given for direct products of bipartite graphs to have total chromatic number
equal to its maximum degree plus one. Partial results towards the total
chromatic number of the direct product of complete graphs are also established.
