A polynomial embedding of pairs of orthogonal partial latin squares

TL;DR

This paper presents the first direct polynomial order embedding method for pairs of orthogonal partial Latin squares, showing they can be embedded into larger orthogonal Latin squares with order bounds related to the original size.

Contribution

It introduces a novel polynomial order embedding construction for orthogonal partial Latin squares, improving previous understanding of their embedding capabilities.

Findings

Pairs of orthogonal partial Latin squares can be embedded in orthogonal Latin squares of order at most 16n^4.

All orders greater than or equal to 48n^4 are sufficient for such embeddings.

This is the first direct polynomial order embedding construction in the literature.

Abstract

We show that a pair of orthogonal partial latin squares of order $n$ can be embedded in a pair of orthogonal latin squares of order at most $16n^4$ and all orders greater than or equal to $48n^4$. This paper provides the first direct polynomial order embedding construction in the literature.