On multipartite derangement graphs

TL;DR

This paper explores the structure of derangement graphs of transitive permutation groups, introducing new families with complete multipartite graphs and analyzing independence numbers for groups of degree 2p.

Contribution

It presents two new families of transitive groups with complete multipartite derangement graphs and bounds the independence number for groups of degree 2p.

Findings

Identified new families of transitive groups with complete multipartite derangement graphs.

Proved an upper bound on the independence number for groups of degree 2p.

Extended understanding of the structure of derangement graphs in permutation groups.

Abstract

Given a finite transitive permutation group $G\leq \operatorname{Sym}(\Omega)$, with $|\Omega|\geq 2$, the derangement graph $\Gamma_G$ of $G$ is the Cayley graph $\operatorname{Cay}(G,\operatorname{Der}(G))$, where $\operatorname{Der}(G)$ is the set of all derangements of $G$. Meagher et al. [On triangles in derangement graphs, {\it J. Combin. Theory Ser. A}, 180:105390, 2021] recently proved that $\operatorname{Sym}(2)$ acting on $\{1,2\}$ is the only transitive group whose derangement graph is bipartite and any transitive group of degree at least three has a triangle in its derangement graph. They also showed that there exist transitive groups whose derangement graphs are complete multipartite. This paper gives two new families of transitive groups with complete multipartite derangement graphs. In addition, we prove that if $p$ is an odd prime and $G$ is a transitive group of degree $2p$, then the independence number of $\Gamma_{G}$ is at most twice the size of a point-stabilizer of $G$.