A construction for biembeddings of Latin squares (Q640450)

The article [\textit{M. J.\ Grannell, T. S.\ Griggs} and \textit{J.\ Širáň}, ``Recursive constructions for triangulations'', J. Graph Theory 39, No.2, 87--107 (2002; Zbl 0999.05021)] contains a construction of non-isomorphic face \(2\)-colorable triangular embeddings of \(K_{n,n,n}\) in closed \(2\)-manifolds for certain values of \(n\); this is an important component in establishing the best known lower bound for the number of non-isomorphic triangular embeddings of \(K_n\) for these \(n\). In the current article the authors generalize this construction (finishing a work begun in [\textit{D. M.\ Donovan, A.\ Drápal, M. J.\ Grannell, T. S.\ Griggs} and \textit{J. G.\ Lefevre}, ``Quarter-regular biembeddings of Latin squares'', Discrete Math. 310, No. 4, 692--699 (2010; Zbl 1228.05085)]) and investigate the family of underlying latin squares. This new construction eliminates the need for a parallel class in one of the colors. Using the new construction the authors present an infinite class of biembeddings, none of which can be obtained from the previous construction.