Lifting Graph Automorphisms by Voltage Assignments

Abstract

The problem of lifting graph automorphisms along covering projections and the analysis of lifted groups is considered in a purely combinatorial setting. The main tools employed are: (1) a systematic use of the fundamental groupoid; (2) unification of ordinary, relative and permutation voltage constructions into the concept of a voltage space; (3) various kinds of invariance of voltage spaces relative to automorphism groups; and (4) investigation of geometry of the lifted actions by means of transversals over a localization set. Some applications of these results to regular maps on surfaces are given. Because of certain natural applications and greater generality, graphs are allowed to have semiedges. This requires careful re-examination of the whole subject and at the same time leads to simplification and generalization of several known results.

Some recent discoveries about half-arc-transitive graphs

“…We can now describe a certain voltage assignment of the voltages from J to the set of the arcs of DC 8 . (For the terminology on graph coverings via voltage assignments, see [13].) For each i ∈ Z 8 , we put the trivial voltage 1 on all of the arcs corresponding to the edge f i , and put the voltage ϕ i (a) on the arc corresponding to e i (directed from i to i + 1).…”

О теоремax Оикавы и Аракавы для графов

“…Following [7] we define a graph with semi-edges as an ordered quadruple X = (D, V ; I, λ) where D = D(X) is a set of darts, V = V (X) is a nonempty set of vertices, which is required to be disjoint from D, I is a mapping of D onto V, called the incidence function, and λ is an involutory permutation of D, called the dart-reversing involution. For convenience or if λ is not explicitly specified we sometimes writex instead of λx.…”

“…We will refer to them as semi-edges or the tails of graph (X/G) tail . For the basic facts of theory of graphs with semiedges see Section 2 and the papers ( [7], [3], [4]). Denote genus of (X/G) tail by g(X/G) tail .…”

“…The second way is to consider the images of invertible edges as semi-edges (or tails) of the factor graph X/G = (X/G) tail , and the third one is to create the factor graph X/G = (X/G) f ree by deleting loops (or semi-edges) that are the images of invertible edges. All the three ways are well known in the literature ( [7], [3], [2]); they are effectively used in various questions of the graph theory. Figure 1.…”

Tetravalent one-regular graphs of order 4p2

“…The next proposition is a special case of [30,Theorem 4.2]. For more results on graph covers we refer the reader to [1,2,14,28,29].…”