Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  655.05018
Autor:  Erdös, Paul; Hickerson, D.R.; Norton, D.A.; Stein, S.K.
Title:  Has every Latin square of order n a partial Latin transversal of size n-1? (In English)
Source:  Am. Math. Mon. 95, No.5, 428-430 (1988).
Review:  The authors have two conjectures:
Conjecture 1: An equi-n-square has a transversal with at least n-1 distinct symbols. [For definition of equi-n-square, see S. Stein, Transversals of Latin squares and their generalizations, Pac. J. Math. 59, 567-575 (1975; Zbl 302.05015)].
Conjecture 2: An n-1 by n array in which each symbol appears at most q times (q \leq n) has a Latin transversal.
The authors' definition of a Latin transversal is the same as the usual definition of a transversal, (see e.g. the reviewer's joint book with A. D. Keedwell [Latin squares (1974; Zbl 283.05014)]).
{Reviewer's remarks:
1.) I. I. Derienko published a paper [Mat. Issled. 102, 53-65 (1988)] in which he proved a weaker version of Conjecture 1. Namely that every Latin square has a transversal with at least n-1 distinct symbols.
The reviewer was not able to understand Derienko's proof. At present some outstanding mathematicians of the field are checking Derienko's proof.
2.) There is a comment by the Editor which is not quite accurate. The Editor claims that the reviewer made a conjecture equivalent to Conjecture 1 [see Research Problem 40. Period. Math. Hung. 17, 245-246 (1986; Zbl 609.05018)], the purpose of the reviewer's paper was: knowing Stein's original conjecture to make a more general conjecture whose validity implies Conjecture 1.}
Reviewer:  J.Denes
Classif.:  * 05B15 Orthogonal arrays, etc.
                   00A07 Problem books
Keywords:  Latin transversal
Citations:  Zbl 315.05012; Zbl 302.05015; Zbl 283.05014; Zbl 609.05018

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