Hamiltonicity for monolithic groups with cyclic quotient

Establish that, for every finite monolithic group G with non-abelian monolith N=S^k, where S is a non-abelian finite simple group and G/N is cyclic, the generating graph Γ(G) is Hamiltonian when S is sufficiently large; determine whether the weaker hypothesis that G, equivalently N, is sufficiently large also suffices.

Background

The paper proves that the generating graph of a sufficiently large almost simple group with Lie-type socle is Hamiltonian whenever the quotient by the socle is cyclic. The proposed extension replaces the simple socle by a non-abelian monolith N=S^k, thereby encompassing a broader class of monolithic groups.

The authors explain that the probabilistic estimate of Harper and Quick extends to monolithic groups, but the pointwise generation estimate used to construct the Hamiltonian cycle does not currently have a suitable analogue for general monolithic groups. They note that the conjecture is a special case of the broader Hamiltonicity conjecture for finite groups whose proper quotients are cyclic, and they leave open whether the stated size condition can be weakened from sufficiently large S to sufficiently large G or N.