Disconnected Cliques in Derangement Graphs

TL;DR

This paper explores the relationship between orthogonal Latin squares and disconnected maximal cliques in derangement graphs, providing modular conditions and a proof related to Euler's 36 Officer Problem.

Contribution

It establishes a novel correspondence between Latin squares and graph cliques, introduces modular obstructions for clique existence, and offers an elementary proof of the nonexistence of Euler's 36 Officer Solution.

Findings

Established a correspondence between Latin squares and graph cliques.

Derived modular conditions for the existence of disconnected maximal cliques.

Provided a new elementary proof of the nonexistence of Euler's 36 Officer Solution.

Abstract

We obtain a correspondence between pairs of $N\times N$ orthogonal Latin squares and pairs of disconnected maximal cliques in the derangement graph with $N$ symbols. Motivated by methods in spectral clustering, we also obtain modular conditions on fixed point counts of certain permutation sums for the existence of collections of mutually disconnected maximal cliques. We use these modular obstructions to analyze the structure of maximal cliques in $X_N$ for small values of $N$. We culminate in a short, elementary proof of the nonexistence of a solution to Euler's $36$ Officer Problem.