Forbidden subgraphs in generating graphs of finite groups

Abstract

Let G be a 2-generated finite group. The generating graph Γ( G ) is the graph whose vertices are the elements of G and where two vertices g 1 and g 2 are adjacent if G = h g 1 ,g 2 i . This graph encodes the combinatorial structure of the distribution of generating pairs across G. In this paper we study some graph theoretic properties of Γ( 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 Γ( 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 other things, that Γ( S n ) and Γ( A n ) are perfect if and only if n (cid:54) 4). Finally we prove that for a finite group G , the properties that Γ( G ) is split, chordal or C 4 -free are equivalent.