A Construction for Biembeddings of Latin Squares

An existing construction for face 2-colourable triangular embeddings of complete regular tripartite graphs is extended and then re-examined from the viewpoint of the underlying Latin squares. We prove that this generalization gives embeddings which are not isomorphic to any of those produced by the original construction.

[1]  Jonathan L. Gross,et al.  Topological Graph Theory , 1987, Handbook of Graph Theory.

[2]  Mike J. Grannell,et al.  A lower bound for the number of triangular embeddings of some complete graphs and complete regular tripartite graphs , 2008, J. Comb. Theory, Ser. B.

[3]  Seiya Negami,et al.  Three nonisomorphic triangulations of an orientable surface with the same complete graph , 1994, Discret. Math..

[4]  G. Ringel.  Map Color Theorem , 1974 .

[5]  Mike J. Grannell,et al.  A lower bound for the number of orientable triangular embeddings of some complete graphs , 2010, J. Comb. Theory, Ser. B.

[6]  Mike J. Grannell,et al.  Exponential Families of Non-Isomorphic Triangulations of Complete Graphs , 2000, J. Comb. Theory, Ser. B.

[7]  Vladimir P. Korzhik.  Exponentially many nonisomorphic orientable triangular embeddings of K12s , 2008, Discret. Math..

[8]  Mike J. Grannell,et al.  Quarter-regular biembeddings of Latin squares , 2010, Discret. Math..

[9]  Mike J. Grannell,et al.  BIEMBEDDINGS OF LATIN SQUARES AND HAMILTONIAN DECOMPOSITIONS , 2004, Glasgow Mathematical Journal.

[10]  Mike J. Grannell,et al.  Triangulations of orientable surfaces by complete tripartite graphs , 2006, Discret. Math..

[11]  Mike J. Grannell,et al.  On Biembeddings of Latin Squares , 2009, Electron. J. Comb..

[12]  Vladimir P. Korzhik,et al.  Exponential families of nonisomorphic nonorientable genus embeddings of complete graphs , 2004, J. Comb. Theory, Ser. B.

[13]  Mike J. Grannell,et al.  Recursive constructions for triangulations , 2002 .

[14]  Vladimir P. Korzhik,et al.  On the Number of Nonisomorphic Orientable Regular Embeddings of Complete Graphs , 2001, J. Comb. Theory, Ser. B.

[15]  C. Colbourn,et al.  The CRC handbook of combinatorial designs , edited by Charles J. Colbourn and Jeffrey H. Dinitz. Pp. 784. $89.95. 1996. ISBN 0-8493-8948-8 (CRC). , 1997, The Mathematical Gazette.

[16]  Vladimir P. Korzhik,et al.  Exponentially many nonisomorphic orientable triangular embeddings of K12s+3 , 2009, Discret. Math..

[17]  Vladimir P. Korzhik,et al.  Exponential Families of Non-isomorphic Non-triangular Orientable Genus Embeddings of Complete Graphs , 2002, J. Comb. Theory, Ser. B.