1 1 O ct 2 00 4 Self - avoiding walks and trails on the 3 . 12 2 lattice Anthony

We find the generating function of self-avoiding walks and trails on a semi-regular lattice called the 3.122 lattice in terms of the generating functions of simple graphs, such as self-avoiding walks, polygons and tadpole graphs on the hexagonal lattice. Since the growth constant for these graphs is known on the hexagonal lattice we can find the growth constant for both walks and trails on the 3.122 lattice. A result of Watson [13] then allows us to find the generating function and growth constant of neighbour-avoiding walks on the covering lattice of the 3.122 lattice which is tetravalent. A mapping into walks on the covering lattice allows us to obtain improved bounds on the growth constant for a range of lattices.