On the Existence of Partitioned Incomplete Latin Squares with Five Parts

Authors

Jaromy Kuhl
Donald McGinn
Michael W. Schroeder, Marshall UniversityFollow

Document Type

Article

Publication Date

2019

Abstract

Let a, b, c, d, e be positive integers such that a ≤ b ≤ c ≤ d ≤ e. Heinrich showed the existence of a partitioned incomplete Latin square (PILS) of type (a, b, c) and (a, b, c, d) if and only if a = b = c and 2a ≥ d. For PILS of type (a,b,c,d,e), it is necessary that a + b + c ≥ e, but not sufficient, and no characterization is currently known. In this talk we provide an additional necessary condition, classify the existence of PILS of type (a, b, c, d, a + b + c) and PILS with three equal parts, and show the existence of a family of PILS in which the parts are nearly the same size.

Comments

The copy of record is available from the publisher at https://ajc.maths.uq.edu.au/pdf/74/ajc_v74_p046.pdf. Copyright © 2019 The Authors. Released under the CC-BY-ND 4.0 International License.

Recommended Citation

J. Kuhl, D. G. McGinn and M.W. Schroeder, On the existence of partitioned incomplete Latin squares with five parts. Australas. J. Combin. 74 (2019), 46–60.