A construction for biembeddings of Latin squares
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.
scientific article; zbMATH DE number 5960052