Clique Facets of the Orthogonal Latin Squares Polytope∗

Since 1782, when Euler addressed the question of existence of a pair of Orthogonal Latin Squares (OLS) by stating his famous conjecture ([8, 9, 13]), these structures have remained an active area of research due to their theoretical properties as well as their applications in a variety of fields. In the current work we consider the polyhedral aspects of OLS. In particular we establish the dimension of the OLS polytope, describe all cliques of the underlying intersection graph and categorize them into three classes. For two of these classes we show that the related inequalities have Chvátal rank two and both are facet defining. For each such class, we give a separation algorithm of the lowest possible complexity, i.e. linear in the number of variables.