  A vertex colouring of some graph is called perfect if each vertex of colour
$i$ has exactly $a_{ij}$ neighbours of colour $j$. Being perfect imposes
several restrictions on the colour incidence matrix $(a_{ij})$. We list several
(old and new) necessary conditions for a matrix to be the colour incidence
matrix of a perfect colouring. Moreover we show that a certain combination of
these conditions is also sufficient. Using this we determine a list of all
colour incidence matrices corresponding to perfect colourings of 3-regular,
4-regular and 5-regular graphs with two, three and four colours, respectively.
As an application we determine all perfect colourings of the edge graphs of the
Platonic solids with two, three and four colours, respectively.
