On the flag graphs of regular abstract polytopes: Hamiltonicity and Cayley index

Abstract In this paper, we study the flag graph FG ( P ) of a regular abstract polytope P from two aspects of Cayley graphs: Hamiltonicity and Cayley index. We show that FG ( P ) has a Hamiltonian cycle, and introduce the Cayley index of P as the fraction | A ut ( FG ( P ) ) | ∕ | Γ ( P ) | , where Γ ( P ) is the automorphism group of P . A new construction of arc-transitive tetravalent graphs will be described by means of regular abstract polyhedra of Cayley index larger than 1. In addition, polyhedra of type { p , q } such that p ≤ 5 or q ≤ 5 that have Cayley index larger than 1 are characterized.