Multi-latin squares
Nicholas Cavenagh, Carlo Hamalainen, James G. Lefevre and, Douglas S. Stones

TL;DR
This paper generalizes Latin squares to multi-latin squares, proves embedding theorems, explores their properties such as separability, and connects them to other combinatorial objects, including enumeration for small cases.
Contribution
It introduces multi-latin squares, proves embedding results, demonstrates existence of non-separable cases, and establishes conditions for separability based on the index.
Findings
Any partially filled $k$-latin square can be embedded in a larger one.
Existence of non-separable $k$-latin squares for certain orders.
For large enough $k$, all $k$-latin squares of a given order are separable.
Abstract
A multi-latin square of order and index is an array of multisets, each of cardinality , such that each symbol from a fixed set of size occurs times in each row and times in each column. A multi-latin square of index is also referred to as a -latin square. A -latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square. In this note we show that any partially filled-in -latin square of order embeds in a -latin square of order , for each , thus generalizing Evans' Theorem. Exploiting this result, we show that there exist non-separable -latin squares of order for each . We also show that for each , there exists some finite value such that for all , every -latin square of order is separable. We discuss…
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Taxonomy
Topicsgraph theory and CDMA systems · Graph Labeling and Dimension Problems · Mathematics and Applications
