5 research outputs found
Monomer-dimer tatami tilings of square regions
We prove that the number of monomer-dimer tilings of an square
grid, with monomers in which no four tiles meet at any point is
, when and have the same parity. In addition, we
present a new proof of the result that there are such tilings with
monomers, which divides the tilings into classes of size . The
sum of these tilings over all monomer counts has the closed form
and, curiously, this is equal to the sum of the squares of
all parts in all compositions of . We also describe two algorithms and a
Gray code ordering for generating the tilings with monomers,
which are both based on our new proof.Comment: Expanded conference proceedings: A. Erickson, M. Schurch, Enumerating
tatami mat arrangements of square grids, in: 22nd International Workshop on
Combinatorial Al- gorithms (IWOCA), volume 7056 of Lecture Notes in Computer
Science (LNCS), Springer Berlin / Heidelberg, 2011, p. 12 pages. More on
Tatami tilings at
http://alejandroerickson.com/joomla/tatami-blog/collected-resource
Domino Tatami Covering is NP-complete
A covering with dominoes of a rectilinear region is called \emph{tatami} if
no four dominoes meet at any point. We describe a reduction from planar 3SAT to
Domino Tatami Covering. As a consequence it is NP-complete to decide whether
there is a perfect matching of a graph that meets every 4-cycle, even if the
graph is restricted to be an induced subgraph of the grid-graph. The gadgets
used in the reduction were discovered with the help of a SAT-solver.Comment: 10 pages, accepted at The International Workshop on Combinatorial
Algorithms (IWOCA) 201