On 2-fold covers of graphs
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 canonical 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 A"1"2"k"+"1"0 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.