Forbidden subgraphs in generating graphs of finite groups
Algebraic Combinatorics, Volume 5 (2022) no. 5, pp. 925-946.

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=g 1 ,g 2 . 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 n4). Finally we prove that for a finite group G, the properties that Γ(G) is split, chordal or C 4 -free are equivalent.

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DOI: 10.5802/alco.229
Classification: 20D60,  05C25
Keywords: cographs, generating graph, perfect graphs
Lucchini, Andrea 1; Nemmi, Daniele 1

1 Dipartimento di Matematica “Tullio Levi-Civita” Università degli Studi di Padova 35121 Padova Italy
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Lucchini, Andrea; Nemmi, Daniele. Forbidden subgraphs in generating graphs of finite groups. Algebraic Combinatorics, Volume 5 (2022) no. 5, pp. 925-946. doi : 10.5802/alco.229. https://alco.centre-mersenne.org/articles/10.5802/alco.229/

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