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• W2952629191 abstract "A regular covering projection $pcolon tX to X$ of connected graphs is $G$-admissible if $G$ lifts along $p$. Denote by $tG$ the lifted group, and let $CT(p)$ be the group of covering transformations. The projection is called $G$-split whenever the extension $CT(p) to tG to G$ splits. In this paper, split 2-covers are considered. Supposing that $G$ is transitive on $X$, a $G$-split cover is said to be $G$-split-transitive if all complements $bG cong G$ of $CT(p)$ within $tG$ are transitive on $tX$; it is said to be $G$-split-sectional whenever for each complement $bG$ there exists a $bG$-invariant section of $p$; 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. For cubic symmetric graphs split 2-cover are necessarily cannonical double covers when $G$ is 1- 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 $bG$ is given, and an infinite family of split-transitive 2-covers based on the alternating groups of the form $A_{12k+10}$ 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." @default.
• W2952629191 created "2019-06-27" @default.
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• W2952629191 date "2007-01-25" @default.
• W2952629191 modified "2023-09-27" @default.
• W2952629191 title "On 2-Fold Covers of Graphs" @default.
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