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

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).