On 2-fold covers of graphs

A regular covering projection ℘: X̃ → X of connected graphs is G-admissible if G lifts along ℘. Denote by G̃ the lifted group, and let CT(℘) be the group of covering transformations. The projection is called G-split whenever the extension CT(℘) → G̃ → 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 Ḡ ∼= G of CT(℘) within G̃ are transitive on X̃ ; it is said to be G-split-sectional whenever for each complement Ḡ there exists a Ḡ-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 Ḡ 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.