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Timothy Chow
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There are also "pan-Hamiltonian" Latin squares, see herePerfect Factorisations of Bipartite Graphs and Latin Squares Without Proper Subrectangles by I. M. Wanless, Electronic J. Combin. 6 (1999), R9.

There are also "pan-Hamiltonian" Latin squares, see here.

There are also "pan-Hamiltonian" Latin squares, see Perfect Factorisations of Bipartite Graphs and Latin Squares Without Proper Subrectangles by I. M. Wanless, Electronic J. Combin. 6 (1999), R9.

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Brendan McKay
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There are also "pan-Hamiltonian" Latin squares, see here.