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 Kn,n for n≤9, and count the Latin squares of order 9 without proper subsquares.

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