A Generalization of Plexes of Latin Squares

TL;DR

This paper introduces the concept of k-weights in Latin squares, generalizing the idea of k-plexes, and demonstrates that many non-existence results and conjectures for plexes extend to these weights.

Contribution

It generalizes the concept of k-plexes to k-weights in Latin squares and shows that several existence conjectures for plexes also hold for k-weights.

Findings

Non-existence results for k-plexes extend to k-weights.

Weight-analogues of existence conjectures hold for k-weights.

Provides a unified framework for understanding plexes and weights in Latin squares.

Abstract

A $k$-plex of a latin square is a collection of cells representing each row, column, and symbol precisely $k$ times. The classic case of $k=1$ is more commonly known as a transversal. We introduce the concept of a $k$-weight, an integral weight function on the cells of a latin square whose row, column, and symbol sums are all $k$. We then show that several non-existence results about $k$-plexes can been seen as more general facts about $k$-weights and that the weight-analogues of several well-known existence conjectures for plexes actually hold for $k$-weights.