On the completability of incomplete Latin squares

We introduce the notion of an availability matrix and apply a theorem of Frobenius–König to obtain necessary and sufficient conditions for the completability of an incomplete Latin row. We consider the related problem for two such rows within the framework of (1,2)(1,2)-permutations and give solutions for several special cases. We also show how to extend these results to more than two rows. Finally, we present an integer programming formulation together with polyhedral results, and we discuss some consequences for class-teacher time-table problems.