Latin Squares with Self-Orthogonal Conjugates Original Research Article

In this paper, we investigate the existence of idempotent Latin squares for which each conjugate is orthogonal to precisely its own transpose. We show that the spectrum of Latin squares with this desired property contains all integers v⩾8, with the possible exception of 10 and 11. As an application of our results, it is shown that for all integers v⩾8, with the possible exception of 10 and 11, there exists an idempotent Latin square of order v that realizes the one-regular graph on 6 vertices as a conjugate orthogonal Latin square graph.