Latin squares from multiplication tables. Issue 5 (25th January 2021)

• Record Type: Journal Article
• Authors: Dębski, Michał; Grytczuk, Jarosław
• Is Part Of: Journal of combinatorial designs. Volume 29:Issue 5(2021)
• Page Start: 331
• Page End: 336
• Publication Date: 2021-01-25
• Subjects: Graham's greatest common divisor problem -- Latin square; Combinatorial designs and configurations -- Periodicals; Configurations et schémas combinatoires -- Périodiques; 511.6
• DOI: 10.1002/jcd.21769
• Languages: English
• ISSNs: 1063-8539

Abstract

Abstract: Consider the usual multiplication table n × n in which all entries strictly greater than n were deleted. Can we fill in the empty places so that the resulting table will become a multiplication table of a group? This intriguing question arose in connection to the famous Graham's Greatest Common Divisor Problem. We prove here a weaker statement, namely, that one may always complete such a partial multiplication table to a Latin square. Our result is more general, as it guarantees the possibility of completion of any partial Latin square, provided that the shape of the empty part satisfies certain conditions. We also point on connections of our result to a far‐reaching extension of Graham's problem involving graphs and homogeneous arithmetic progressions.