Two new symmetric inseparable double squares

Two new symmetric inseparable double squares

DISCRETE MATHEMATICS ELSEVIER Discrete Mathematics 141 (1995) 275 277 Note Two new symmetric inseparable double squares Xiaomin Bao* Department of ...

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DISCRETE MATHEMATICS ELSEVIER

Discrete Mathematics 141 (1995) 275 277

Note

Two new symmetric inseparable double squares Xiaomin Bao* Department of Mathematics and Astro~ omy. University 0[ Manitoba. Winnipeg, Manitoba. Conada R3T 2N2

Received 29 October 1992; revised 19 August 1993

Abstract

In this paper we show the existence of an SIDS (18) and an SIDS (23), which allows us to completely prove Mouyart's conjecture.

A symmetric double square of order n based on the n-set R is an n x n array whose cells contain unordered pairs of symbols of R, such that each symbol occurs precisely twice per row and twice per column, each pair of distinct elements occurs precisely twice in the array, each pair of identical elements occurs precisely once in the array, and the cells (i,j) and 0", i) contain the same pair. If a symmetric double square can be obtained by superimposing two latin squares, we call it a symmetric separable double square; otherwise we call it a symmetric inseparable double square. A symmetric inseparable double square of order n is denoted by SIDS (n). M o u y a r t I-2] conjectured that there exists an SIDS(n) for every integer n >~ 6. M o u y a r t constructed the squares of all orders except 17, 18, 20, 23 and 24. An SIDS (17), an SIDS (20) and an SIDS (24) have been exhibited in E3]. Professor Zhu Lie (personal communication) has constructed an SIDS (23) as follows. Let A be an I S O L S (23) [1]. Superimpose A on its transpose and put an SIDS (7) in the empty order 7 subsquare. The purpose of this paper is to exhibit an SIDS (18), which is shown in Fig. 1. The symbols on which it is based are ({1,2,3,4} X [ 5 , 6 , 7 , 8 } ) U{ W , Z } . Thus M o u y a r t ' s conjecture has been verified.

* Correspondence address: c/o Ms. Li Lin, Xinghua Book Store, Chonqing Distribution Agency, Tanziku, Chongqing 630050, China. 0012-365X/95/$09.50 © 1995-Elsevier Science B.V. All rights reserved SSDI 0 0 1 2 - 3 6 5 X ( 9 4 ) E 0 2 2 4 - R

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Acknowledgement The author would like to thank Professor Zhu Lie for his warm encouragement and help.

References [1] K. Heinrich and L. Zhu, Incomplete self-orthogonal latin squares, J Aust. Math. Soc. Ser. A 42 (1987i 365-384. [2] A.F. Mouyart, Symmetric inseparable double squares, Ann. Discrete Math. 17 (1983) 483 496. [3] Xiaomin Bao, On a Mouyart's conjecture, Discrete Math. 98 (1991) 141 145.