Latin Squares with Self-Orthogonal Conjugates

In this paper, we investigate the existence of idempotent Latin squares for which each conjugate is orthogonal to precisely its own transpose. We show that the spectrum of Latin squares with this desired property contains all integers v>=8, with the possible exception of 10 and 11. As an application of our results, it is shown that for all integers v>=8, with the possible exception of 10 and 11, there exists an idempotent Latin square of order v that realizes the one-regular graph on 6 vertices as a conjugate orthogonal Latin square graph.

[1]  Haim Hanani,et al.  Balanced incomplete block designs and related designs , 1975, Discret. Math..

[2]  Ruizhong Wei,et al.  The existence of Schröder designs with equal-sized holes , 1997, Discret. Math..

[3]  N. S. Mendelsohn.  SELF‐ORTHOGONAL WEISNER DESIGNS , 1979 .

[4]  Alexander Schrijver,et al.  Group divisible designs with block-size four , 2006, Discret. Math..

[5]  Richard M. Wilson,et al.  Constructions and Uses of Pairwise Balanced Designs , 1975 .

[6]  R. D. Baker.  Quasigroups and tactical systems , 1978 .

[7]  F. E. Bennett,et al.  Pairwise balanced designs with prime power block sizes exceeding 7 , 1987 .

[8]  N. S. Mendelsohn,et al.  Orthogonal Latin Charles C . Lindner Square Graphs AUBURN , 2006 .

[9]  N. S. Mendelsohn,et al.  On the construction of Schroeder quasigroups , 1980, Discret. Math..

[10]  Charles C. Lindner,et al.  On the number of conjugates of a quasigroup , 1975 .

[11]  S. Stein.  On the foundations of quasigroups , 1957 .

[12]  Frank E. Bennett.  Latin squares with pairwise orthogonal conjugates , 1981, Discret. Math..

[13]  J. Dénes,et al.  Latin squares and their applications , 1974 .

[14]  Frank E. Bennett,et al.  Incomplete idempotent Schröder quasigroups and related packing designs , 1994 .

[15]  Zhu Lie,et al.  Existence of frame sols of type 2nu1 , 1995 .

[16]  Charles J. Colbourn,et al.  Edge-coloured designs with block size four , 1988 .

[17]  C. Colbourn,et al.  CRC Handbook of Combinatorial Designs , 1996 .