Two Lower Bounds for Self-Assemblies at Temperature 1

Using the Tile Assembly Model proposed by Rothemund and Winfree, we give two lower bounds on the minimum number of tile types needed to uniquely assemble a shape at temperature 1 under a natural assumption that there are no binding domain mismatches (any two adjacent tiles either form a bond or else both touching sides of the tiles are without glues). Rothemund and Winfree showed that uniquely assembling a full N times N square (a square where there is a bond between any two adjacent tiles) at temperature 1 requires N² distinct tile types, and conjectured that the minimum number of tile types needed to self-assemble an N times N square (not a full square) is 2N - 1. Our lower bounds imply that a tile system that uniquely assembles an N times N square without binding domains mismatches, requires at least 2N - 1 tile types.