Upper Bounds on Sets of Orthogonal Colorings of Graphs

TL;DR

This paper extends the concept of orthogonal Latin squares to graph colorings, establishing bounds on the maximum size of sets of orthogonal colorings and unifying various Latin square bounds.

Contribution

It introduces a generalized framework for orthogonal graph colorings, connecting classical Latin square bounds to graph theory and expanding the scope of orthogonal structures.

Findings

Unified bounds for orthogonal Latin squares and graph colorings

Generalized the concept of orthogonality to graph colorings

Established maximum sizes of orthogonal coloring sets

Abstract

We generalize the notion of orthogonal latin squares to colorings of simple graphs. Two $n$-colorings of a graph are said to be \emph{orthogonal} if whenever two vertices share a color in one coloring they have distinct colors in the other coloring. We show that the usual bounds on the maximum size of a certain set of orthogonal latin structures such as latin squares, row latin squares, equi-$n$ squares, single diagonal latin squares, double diagonal latin squares, or sudoku squares are a special cases of bounds on orthogonal colorings of graphs.