Isomorphisms and automorphisms of graph coverings (Q1185095)

A graph \(H\) is called an \(r\)-fold covering of the graph \(G\), if there is an \(r\)-to-one mapping \(p\) (called the covering projection) of the vertex set \(V(H)\) of \(H\) onto the vertex set \(V(G)\) of \(G\) such that the neighbours of each vertex \(v\in V(H)\) are mapped by \(p\) bijectively onto the neighbours of \(p(v)\in V(G)\). If \(\Gamma\) is a subgroup of the automorphism group of \(G\), then a \(\Gamma\)-isomorphism of covering projections \(p:H\to G\) and \(\overline p:\overline H\to G\) is a pair \((\varphi,\psi)\) consisting of an automorphism \(\varphi\in\Gamma\) and an isomorphism \(\psi:H\to\overline H\) such that \(\varphi p=\overline p\psi\). These concepts are studied in the paper together with further concepts, e.g. permutation voltage assignments and lifting automorphisms. At the end also labeled graphs are considered.