  In this paper, perfect k-orthogonal colourings of tensor graphs are studied.
First, the problem of determining if a given graph has a perfect 2-orthogonal
colouring is reformulated as a tensor subgraph problem. Then, it is shown that
if two graphs have a perfect $k$-orthogonal colouring, then so does their
tensor graph. This provides an upper bound on the $k$-orthogonal chromatic
number for general tensor graphs. Lastly, two other conditions for a tensor
graph to have a perfect $k$-orthogonal colouring are given.
