PreprintCC BY-NC-ND 4.0arXiv
On the Hamiltonicity of generating graphs of almost simple groups
Abstract
The generating graph Γ(G) of a finite group G has vertex set G∖{1}, and two distinct vertices are adjacent if and only if they generate G. Breuer, Guralnick, Lucchini, Maroti and Nagy [Bull. Lond. Math. Soc. 42 (2010), 621--633] conjectured that, for every finite group G with at least four elements, Γ(G) contains a Hamiltonian cycle if and only if every proper quotient of G is cyclic. They proved their conjecture for sufficiently large almost simple groups with alternating socle and for all almost simple groups with sporadic socle. In this paper, we complete the asymptotic picture for almost simple groups by proving the conjecture for sufficiently large almost simple groups with socle of Lie type.
Related
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- Uniform Intersection Numbers and Subgroup-Based Generating Graphs of Groups
BibTeX
@misc{iorio2026hamiltonicity,
title = {{On the Hamiltonicity of generating graphs of almost simple groups}},
author = {Luigi Iorio},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09391},
eprint = {2609.09391},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09391},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On the Hamiltonicity of generating graphs of almost simple groups
AU - Luigi Iorio
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09391
UR - https://arxiv.org/abs/2609.09391
AB - The generating graph Γ(G) of a finite group G has vertex set G∖{1}, and two distinct vertices are adjacent if and only if they generate G. Breuer, Guralnick, Lucchini, Maroti and Nagy [Bull. Lond. Math. Soc. 42 (2010), 621--633] conjectured that, for every finite group G with at least four elements, Γ(G) contains a Hamiltonian cycle if and only if every proper quotient of G is cyclic. They proved their conjecture for sufficiently large almost simple groups with alternating socle and for all almost simple groups with sporadic socle. In this paper, we complete the asymptotic picture for almost simple groups by proving the conjecture for sufficiently large almost simple groups with socle of Lie type.
ER -