Forbidden subgraphs in generating graphs of finite groups

TL;DR

This paper explores the structure of generating graphs of finite groups, focusing on forbidden subgraphs, and characterizes when these graphs are cographs, perfect, split, chordal, or free of certain cycles.

Contribution

It provides a complete characterization of generating graphs' properties related to forbidden subgraphs for various classes of finite groups.

Findings

Generating graphs of soluble groups are cographs under certain conditions.

Generating graphs of nilpotent groups are perfect if and only if the group is small.

For finite groups, split, chordal, and C4-free properties are equivalent.

Abstract

Let $G$ be a $2$-generated group. The generating graph $\Gamma(G)$ is the graph whose vertices are the elements of $G$ and where two vertices $g_1$ and $g_2$ are adjacent if $G = \langle g_1, g_2 \rangle.$ This graph encodes the combinatorial structure of the distribution of generating pairs across $G.$ In this paper we study some graph theoretic properties of $\Gamma(G)$, with particular emphasis on those properties that can be formulated in terms of forbidden induced subgraphs. In particular we investigate when the generating graph $\Gamma(G)$ is a cograph (giving a complete description when $G$ is soluble) and when it is perfect (giving a complete description when $G$ is nilpotent and proving, among the others, that $\Gamma(S_n)$ and $\Gamma(A_n)$ are perfect if and only if $n\leq 4$). Finally we prove that for a finite group $G$, the properties that $\Gamma(G)$ is split, chordal or $C_4$-free are equivalent.