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We find the generating function of self-avoiding walks and trails on asemi-regular lattice called the $3.12^2$ lattice in terms of the generatingfunctions of simple graphs, such as self-avoiding walks, polygons and tadpolegraphs on the hexagonal lattice. Since the growth constant for these graphs isknown on the hexagonal lattice we can find the growth constant for both walksand trails on the $3.12^2$ lattice. A result of Watson then allows us to findthe generating function and growth constant of neighbour-avoiding walks on thecovering lattice of the $3.12^2$ lattice which is tetra-valent. A mapping intowalks on the covering lattice allows us to obtain improved bounds on the growthconstant for a range of lattices.Comment: 15 pages, 8 figure

Self-avoiding walks and trails on the 3.122 lattice

Self-avoiding walks and trails on the 3.12² lattice