Quartic Symmetry, Hamiltonicity, and CFSG-Free Structure

The starting point is the use of auxiliary graphs, the so called hexagon graphs, associated with Cayley maps of $(2,s,3)$-groups. Large induced trees in these auxiliary graphs provide a route to Hamilton cycles/paths in the original Cayley graphs. I will then explain why quartic one-regular graphs are central in a possible generalization of this approach to $(2,s,4)$-groups.

In the final part, I will discuss the Feng–Xu normality theorem for quartic Cayley graphs on regular $p$-groups and propose a CFSG-free proof in the special case where one generator has order $p$.

This leads naturally to broader questions about direct proofs in algebraic combinatorics, including primitive groups of degree $2p$ and the Polycirculant conjecture.