A class of orthogonal latin square graphs

An orthogonal latin square graph is a graph whose vertices are latin squares of the same order, adjacency being synonymous with orthogonality. We are interested in orthogonal latin square graphs in which each square is orthogonal to the Cayley table M of a group G and is obtained from M by permuting columns. These permutations, regarded as permutations of G, are orthomorphisms of G and the graphs so obtained are orthomorphism graphs. We will discuss the main problems in the study of orthomorphism graphs and survey bounds on the clique numbers of these graphs. We will also present several problems.

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