几乎单群的生成图的哈密顿性
On the Hamiltonicity of generating graphs of almost simple groups
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中文总结 AI 辅助
本文证明了Breuer等人关于几乎单群生成图哈密顿性的猜想在李型基座情形下成立，从而完成了该猜想的渐近证明。
中文摘要 AI 辅助
有限群$G$的生成图$\Gamma(G)$的顶点集为$G\setminus\{1\}$，两个不同顶点相邻当且仅当它们生成$G$。Breuer、Guralnick、Lucchini、Maroti和Nagy [Bull. Lond. Math. Soc. 42 (2010), 621--633]猜想：对于每个至少含四个元素的有限群$G$，$\Gamma(G)$包含哈密顿圈当且仅当$G$的每个真商群都是循环群。他们证明了该猜想对具有交错基座的足够大的几乎单群以及所有具有散在基座的几乎单群成立。本文通过证明该猜想对具有李型基座的足够大的几乎单群成立，完成了几乎单群的渐近图景。
英文摘要
The generating graph $Γ(G)$ of a finite group $G$ has vertex set $G\setminus\{1\}$, 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.
