The problem of lifting graph automorphisms along covering projections is

Regular homomorphisms of oriented maps essentially arise from a factorization by a subgroup of automorphisms. This kind of map homomorphisms is studied in detail, and generalized to the case when the induced homomorphism of the underlying graphs is not valency preserving. Reconstruction is treated by means of voltage assignments on angles, a natural extension of the common assignments on darts. Lifting and projecting groups of automorphisms along regular homomorphisms is studied in some detail. Finally, the split-extension structure of lifted groups is analyzed.

An action graph is a combinatorial representation of a group acting on a set. Comparing two group actions by a morphism of actions induces a covering projection of the respective graphs. This simple observation generalizes and unies many well-known results in graph theory, with applications ranging from the theory of maps on surfaces and group presentations to theoretical computer science, among others. Reconstruction of action graphs from smaller ones is considered, some results on lifting and projecting the equivariant group of automorphisms are proved, and a special case of the split-extension structure of lifted groups is studied. Action digraphs in connection with group presentations are also discussed. 1

A regular covering projection ℘: ˜ X → X of connected graphs is G-admissible if G lifts along ℘. Denote by ˜ G the lifted group, and let CT(℘) be the group of covering transformations. The projection is called G-split whenever the extension CT(℘) → ˜G → G splits. In this paper, split 2-covers are considered, with a particular emphasis given to cubic symmetric graphs. Supposing that G is transitive on X, a G-split cover is said to be G-split-transitive if all complements ¯ G ∼ = G of CT(℘) within ˜ G are transitive on ˜ X; it is said to be G-split-sectional whenever for each complement ¯ G there exists a ¯ G-invariant section of ℘; and it is called G-split-mixed otherwise. It is shown, when G is an arc-transitive group, split-sectional and split-mixed 2-covers lead to canonical double covers. Split-transitive covers, however, are considerably more difficult to analyze. For cubic symmetric graphs split 2-cover are necessarily cannonical double covers (that is, no G-split-transitive 2-covers exist) when G is 1-regular or 4-regular. In all other cases, that is, if G is s-regular, s = 2, 3 or 5, a necessary and sufficient condition for the existence of a transitive complement ¯G is given, and moreover, an infinite family of split-transitive 2-covers based on the alternating groups of the form A12k+10 is constructed. Finally, chains of consecutive 2-covers, along which an arc-transitive group G has successive lifts, are also considered. It is proved that in such a chain, at most two projections can be split. Further, it is shown that, in the context of cubic symmetric graphs, if exactly two of them are split, then one is split-transitive and the other one is either split-sectional or split-mixed.