2007/01/25 by Yan‐Quan Feng, Yan-Quan Feng, Klavdija Kutnar +8 · 1 citation
Engineering · Mathematics · #05C25 #57M15 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #graph theory and CDMA systems #math.CO #math.GR #msc:05C25 #msc:57M15
paper · pdf · doi:10.48550/arxiv.math/0701722
18 pages, 3 figures
arxiv created 2007/01/25 · openalex publication_date 2007/01/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A regular covering projection \p\colon \tX → 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) → \tG → 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 ≅ 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 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.