On triangles in derangement graphs

TL;DR

This paper proves that derangement graphs of transitive permutation groups of degree at least 3 always contain triangles, and explores the maximum ratio of independence number to stabilizer size.

Contribution

It establishes the existence of triangles in derangement graphs for all transitive groups of degree at least 3 and analyzes the extremal ratio of independence number to stabilizer size.

Findings

Derangement graphs of transitive groups of degree ≥ 3 always contain triangles.

Examples of groups where the independence number to stabilizer size ratio is maximized.

Insights into the structure of derangement graphs and their independence numbers.

Abstract

Given a permutation group $G$, the derangement graph $\Gamma_G$ of $G$ is the Cayley graph with connection set the set of all derangements of $G$. We prove that, when $G$ is transitive of degree at least $3$, $\Gamma_G$ contains a triangle. The motivation for this work is the question of how large can be the ratio of the independence number of $\Gamma_G$ to the size of the stabilizer of a point in $G$. We give examples of transitive groups where this ratio is maximum.