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

摘要

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 vertical bar Aut(FG(P))vertical bar/vertical bar Gamma(P)vertical bar, where Gamma(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. (C) 2019 Elsevier B.V. All rights reserved.