The clone cover

Abstract

Each finite graph on n vertices determines a special (n − 1)-fold covering graph that we call the clone cover. Several equivalent definitions and basic properties about this remarkable construction are presented. In particular, we show that for k ≥ 2, the clone cover of a k-connected graph is k-connected, the clone cover of a planar graph is planar and the clone cover of a hamiltonian graph is hamiltonian. As for symmetry properties, in most cases we also understand the structure of the automorphism groups of these covers. A particularly nice property is that every automorphism of the base graph lifts to an automorphism of its clone cover. We also show that the covering projection from the clone cover onto its corresponding 2-connected base graph is never a regular covering, except when the base graph is a cycle.