# Perfect factorisations of bipartite graphs and Latin squares without proper subrectangles

A Latin square is pan-Hamiltonian if every pair of rows forms a single
cycle. Such squares are related to perfect 1-factorisations of the
complete bipartite graph. A square is atomic if every conjugate is
pan-Hamiltonian. These squares are indivisible in a strong sense --
they have no proper subrectangles. We give some existence results and
a catalogue for small orders. In the process we identify all the
perfect 1-factorisations of K_{n,n} for n≤9, and count the
Latin squares of order 9 without proper subsquares.
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