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The number of solutions to an equation arising from a problem on latin squares

Part of: Designs and configurations

Published online by Cambridge University Press:  09 April 2009

I. P. Goulden  and
S. A. Vanstone
I. P. Goulden
Affiliation:
University of WaterlooWaterloo, Ontario, Canada
S. A. Vanstone
Affiliation:
University of WaterlooWaterloo, Ontario, Canada
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Abstract

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A recent article of G. Chang shows that an n × n partial latin square with prescribed diagonal can always be embedded in an n × n latin square except in one obvious case where it cannot be done. Chang's proof is to show that the symbols of the partial latin square can be assigned the elements of the additive abelian group Zn so that the diagonal elements of the square sum to zero. A theorem of M. Halls then shows this to be embeddable in the operation table of the group. In this paper, we show that when n is a prime one can determine exactly the number of distinct ways in which this assignment can be made. The proof uses some graph theoretic techniques.

MSC classification

Secondary: 05B15: Orthogonal arrays, Latin squares, Room squares
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 34 , Issue 1 , February 1983 , pp. 138 - 142
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Bondy, J. A. and Murty, U. S. R., Graph theory with applications (Amer. Elsevier, 1976).CrossRefGoogle Scholar
[2]Chang, Gerald J., ‘Complete diagonals of latin squares’, Canad. Math. Bull. 22 (1979), 477–481.CrossRefGoogle Scholar
[3]Gilbert, E. N., ‘Enumeration of labelled graphs’, Canad. J. Math. 8 (1956), 405–411.CrossRefGoogle Scholar
[4]Hall, M. Jr, ‘A combinatorial problem on abelian groups’, Proc. Amer. Math. Soc. 3 (1952), 584–587.CrossRefGoogle Scholar