Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: On biembeddings of Latin squares. | On biembeddings of Latin squares M. J. Grannell T. S. Griggs Department of Mathematics and Statistics The Open University Walton Hall Milton Keynes MK7 6AA United Kingdom M. Knor Department of Mathematics Faculty of Civil Engineering Slovak University of Technology Radlinskeho 11 813 68 Bratislava Slovakia Submitted Sep 7 2008 Accepted Aug 10 2009 Published Aug 21 2009 Mathematics Subject Classifications 05B15 05C10 Abstract A known construction for face 2-colourable triangular embeddings of complete regular tripartite graphs is re-examined from the viewpoint of the underlying Latin squares. This facilitates biembeddings of a wide variety of Latin squares including those formed from the Cayley tables of the elementary Abelian 2-groups C2 k 2 . In turn these biembeddings enable us to increase the best known lower bound for the number of face 2-colourable triangular embeddings of Kn n for an infinite class of values of n. 1 Background In 6 a recursive construction was presented for face 2-colourable triangular embeddings of complete tripartite graphs Kn n n. The construction was used in that paper to provide lower bounds of the form 2an for the numbers of face 2-colourable triangular embeddings of both complete tripartite graphs Knw and complete graphs Kn for certain values of n. In a subsequent paper 2 a generalization of this construction was used to increase these lower bounds to ones of the form nan for certain values of n. A face 2-colourable triangular embedding of Kn n n corresponds to a biembedding of two Latin squares. The purpose of this current paper is to re-examine the construction from 6 from the viewpoint of the Latin squares involved. This alternative focus enables us to obtain new results about biembeddings of Latin squares and to improve the bound given in 2 . For general background material on topological embeddings we refer the reader to 7 and 9 . Our embeddings will always be in closed connected 2-manifolds without a boundary. A graph embedding .
