Perfect precise colourings of uniform tilings with regular star polygons

This paper deals with the problem of colouring n-valent uniform tilings with regular star polygons using n distinct colours, such that one tile of each colour occurs at each vertex (precise colouring), and such that every element of the symmetry group of the tiling induces a permutation of the colours (perfect colouring). A group theoretic and geometric method for enumerating all perfect precise colourings of the given tiling is presented. The 29 uniform tilings with regular star polygons admit precisely 37 perfect precise colourings. Among the 29 tilings, 14 do not admit perfect precise colourings.