An improved upper bound for the \(3\)-dimensional dimer problem
The authors quote various known lower and upper bounds for the number of ``dimer tilings'' of a \(3\)-dimensional cuboid. The known number of dimer tilings of a plane clearly gives a crude lower bound, and several improvements of this are quoted. An upper bound is found in this paper by assuming the brick is divided into layers and considering the situation when \(\ell\) dimers have one end in each layer. A graph \(S\) is defined for each such choice of \(\ell\) dimers, and the number of tilings of the whole is related to the number of closed walks on \(S\) appropriately weighted. An ingenious group theoretical argument then makes it possible to estimate the desired number of tilings. The result is reasonably close to the best known lower bound.
scientific article; zbMATH DE number 1425137
scientific article