  Two colourings of a graph are orthogonal if they have the property that when
two vertices are coloured with the same colour in one colouring, then those
vertices receive distinct colours in the other colouring. In this paper,
orthogonal colourings of Cayley graphs are discussed. Firstly, the orthogonal
chromatic number of cycle graphs are completely determined. Secondly, the
orthogonal chromatic number of certain circulant graphs is explored. Lastly,
orthogonal colourings of product graphs and Hamming graphs are studied.
