Journal of Combinatorics

Volume 14 (2023)

Number 1

Completing partial Latin squares with two filled rows and three filled columns

Pages: 139 – 153



Carl Johan Casselgren (Department of Mathematics, Linköping University, Linköping, Sweden)

Herman Göransson (Department of Mathematics, Linköping University, Linköping, Sweden)


Consider a partial Latin square $P$ where the first two rows and first three columns are completely filled, and every other cell of $P$ is empty. It has been conjectured that all such partial Latin squares of order at least $8$ are completable. Based on a technique by Kuhl and McGinn we describe a framework for completing partial Latin squares in this class. Moreover, we use our method for proving that all partial Latin squares from this family, where the intersection of the nonempty rows and columns form a Latin rectangle with three distinct symbols, are completable.


Latin square, partial Latin square, completing partial Latin squares

2010 Mathematics Subject Classification

Primary 05B15. Secondary 05C15.

This paper is based on the Bachelor thesis [7] by Göransson written under the supervision of Casselgren.

Casselgren was supported by a grant from the Swedish Research Council (2017-05077).

Received 10 September 2020

Published 19 August 2022