Ryser's Theorem for Symmetric $\rho$-latin Squares

TL;DR

This paper extends Ryser's theorem to symmetric $ ho$-latin squares, providing necessary and sufficient conditions for their construction with specified diagonal entries, generalizing previous results on Latin squares.

Contribution

It introduces a comprehensive characterization for symmetric $ ho$-latin squares with prescribed diagonal entries, extending prior theorems by Cruse, Goldwasser et al., and Andersen-Hoffman.

Findings

Established necessary and sufficient conditions for symmetric $ ho$-latin squares.

Extended the Andersen-Hoffman Theorem to include specified diagonal entries.

Unified previous results on Latin squares with new symmetry and diagonal constraints.

Abstract

Let $L$ be an $n\times n$ array whose top left $r\times r$ subarray is filled with $k$ different symbols, each occurring at most once in each row and at most once in each column. We establish necessary and sufficient conditions that ensure the remaining cells of $L$ can be filled such that each symbol occurs at most once in each row and at most once in each column, $L$ is symmetric with respect to the main diagonal, and each symbol occurs a prescribed number of times in $L$. The case where the prescribed number of times each symbol occurs is $n$ was solved by Cruse (J. Combin. Theory Ser. A 16 (1974), 18--22), and the case where the top left subarray is $r\times n$ and the symmetry is not required, was settled by Goldwasser et al. (J. Combin. Theory Ser. A 130 (2015), 26--41). Our result allows the entries of the main diagonal to be specified as well, which leads to an extension of the Andersen-Hoffman Theorem (Annals of Disc. Math. 15 (1982) 9--26, European J. Combin. 4 (1983) 33--35).