Isotopy graphs of Latin tableaux

TL;DR

This paper extends the concept of isotopy from Latin squares to Latin tableaux, introduces isotopy graphs, and explores their structure, including the existence of high-dimensional cubes and triangle-free properties.

Contribution

It defines isotopy graphs for Latin tableaux, characterizes their structure, and constructs examples with specific properties such as high-dimensional cubes.

Findings

Existence of Latin tableaux with isotopy graphs as d-dimensional cubes

Most isotopy graphs are triangle-free

Characterization of Latin tableaux with triangle-containing isotopy graphs

Abstract

Latin tableaux are a generalization of Latin squares, which first appeared in the early 2000's in a paper of Chow, Fan, Goemans, and Vondr\'{a}k. Here, we extend the notion of isotopy, a permutation group action, from Latin squares to Latin tableaux. We define isotopy graphs for Latin tableaux, which encode the structure of orbits under the isotopy action, and investigate the relationship between the shape of a Latin tableau and the structure of its isotopy graph. Our main result shows that for any positive integer $d$, there is a Latin tableau whose isotopy graph is a $d$-dimensional cube. We show that most isotopy graphs are triangle-free, and we give a characterization of all the Latin tableaux for which the isotopy graph contains a triangle. We also give a formula for the degree of a vertex of each component of an isotopy graph which depends on both the shape of the Latin Tableaux and the filling.