Pseudo orthogonal Latin squares
The authors prove that the existence of a family of \(m\) mutually pseudo orthogonal Latin squares of order \(n\) (\(\mathrm{POLS}(m;n)\)) is equivalent to the existence of a family of \(m\) mutually orthogonal Latin squares of order \(n\) (\(\mathrm{MOLS}(m;n)\)). Two distinct Latin squares \(A\), \(B\) of order \(n\) are said to be pseudo orthogonal if for each \(1\leq i\), \(j\leq n\) there is a \(k\) with \(1\leq k\leq n\) satisfying \(A(i,k)=B(j,k)\). The notion of pseudo orthogonality was first introduced in [\textit{N. S. Bolshakova}, Discrete Math. Appl. 18, No. 2, 187--197 (2008; Zbl 1200.05141); translation from Diskretn. Mat. 20, No. 1, 70--79 (2008)]. The authors use graph invariants to establish a relationship between pseudo orthogonal Latin squares and orthogonal Latin squares. Specifically, for a graph \(G\), these invariants are the clique partition number of \(G\) and the size \(\pi (G)\) of a prime set for an optimum \(\pi\)-labeling of \(G\). For \(m\geq 0\) and \(n\geq 2\) let \(G\) be the graph obtained by taking the union of \(m+2\) vertex disjoint complete graphs on \(n\) vertices. The authors' theorem asserting the equivalence of a family of \(\mathrm{MOLS}(m;n)\) and a family of \(\mathrm{POLS}(m;n)\) also has two intermediate equivalent statements: (1) \(\pi (G)=n^2\) and (2) there is a clique partition of \(\bar G\) which consists only of \(K_{m+2}\)'s. The authors apply this theorem and surrounding ideas to obtain exact values of clique partition numbers of several classes of complete multipartite graphs and tensor products of complete graphs.
mutually pseudo orthogonal Latin squares
scientific article; zbMATH DE number 7316009