  We consider the problem of counting the number of 3-colourings of the edges
(bonds) of the 4-8 lattice and the 3-12 lattice. These lattices are Archimedean
with coordination number 3, and can be regarded as decorated versions of the
square and honeycomb lattice, respectively. We solve these edge-colouring
problems in the infinite-lattice limit by mapping them to other models whose
solution is known. The colouring problem on the 4-8 lattice is mapped to a
completely packed loop model with loop fugacity n=3 on the square lattice,
which in turn can be mapped to a six-vertex model. The colouring problem on the
3-12 lattice is mapped to the same problem on the honeycomb lattice. The
3-edge-colouring problems on the 4-8 and 3-12 lattices are equivalent to the
3-vertex-colouring problems (and thus to the zero-temperature 3-state
antiferromagnetic Potts model) on the "square kagome" ("squagome") and
"triangular kagome" lattices, respectively.
