Improved bounds on completion of partial Latin squares
Abstract
A Latin square of order n is an n × n array filled with n symbols so that each symbol appears exactly once in every row and column. A partial Latin square of order n is an n × n array whose cells are either empty or filled in such a way that each symbol appears at most once in every row and column, and at most n distinct symbols are used. In 1983, Daykin and Häggkvist conjectured that every partial Latin square in which each row and column contains at most n/4 symbols, and each symbol is used at most n/4 times, can be completed to a Latin square. We prove that every partial Latin square in which each row and column contains at most 0.231n symbols, and each symbol is used at most 0.231n times, can be completed to a Latin square, significantly improving the previous best-known bound of 0.08n, obtained by Fu and Weng. This problem can be seen as a partite analogue of the Nash-Williams conjecture concerning triangle decompositions of dense graphs, recently proved in the breakthrough work of Delcourt and Postle. Our proof uses a `discharging' strategy, adapting the approach of Delcourt and Postle, combined with a novel method to achieve a `balancedness' property, required for the partite setting.
Related
BibTeX
@misc{allsop2026improved,
title = {{Improved bounds on completion of partial Latin squares}},
author = {Jack Allsop and Candida Bowtell and Thomas Lesgourgues and Kalina Petrova},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37877},
eprint = {2609.37877},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37877},
}
RIS
TY - UNPB
TI - Improved bounds on completion of partial Latin squares
AU - Jack Allsop
AU - Candida Bowtell
AU - Thomas Lesgourgues
AU - Kalina Petrova
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37877
UR - https://arxiv.org/abs/2609.37877
AB - A Latin square of order n is an n × n array filled with n symbols so that each symbol appears exactly once in every row and column. A partial Latin square of order n is an n × n array whose cells are either empty or filled in such a way that each symbol appears at most once in every row and column, and at most n distinct symbols are used. In 1983, Daykin and Häggkvist conjectured that every partial Latin square in which each row and column contains at most n/4 symbols, and each symbol is used at most n/4 times, can be completed to a Latin square. We prove that every partial Latin square in which each row and column contains at most 0.231n symbols, and each symbol is used at most 0.231n times, can be completed to a Latin square, significantly improving the previous best-known bound of 0.08n, obtained by Fu and Weng. This problem can be seen as a partite analogue of the Nash-Williams conjecture concerning triangle decompositions of dense graphs, recently proved in the breakthrough work of Delcourt and Postle. Our proof uses a `discharging' strategy, adapting the approach of Delcourt and Postle, combined with a novel method to achieve a `balancedness' property, required for the partite setting.
ER -