On the Hamiltonicity of generating graphs of almost simple groups

The generating graph $\mathrm{\Gamma }(G)$ 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, $\mathrm{\Gamma }(G)$ 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.