Disconnected Cliques in Derangement Graphs

TL;DR

The paper addresses the existence and structure of mutually orthogonal Latin squares (MOLS) by linking them to pairs of disconnected maximal cliques in the derangement graph $X_N$. It develops a bijection between orthogonal Latin square pairs and disconnected clique pairs, and leverages spectral clustering via representation theory to derive modular obstructions that constrain possible clique configurations. Through detailed analysis for $N=3,4,5$, and a concise, elementary nonexistence proof for Euler's $36$ Officer Problem ($N=6$), it demonstrates how modular representations yield concrete obstructions and structure theorems. The work introduces a novel algebraic-graph-theoretic framework that sheds light on MOLS existence questions and provides tools that may guide future constructions and impossibility results.

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.

Theorems & Definitions (42)

• Proposition 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5
• proof
• Remark 2.6