| Naslov: | On 2-fold covers of graphs |
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| Avtorji: | ID Feng, Yan-Quan (Avtor)
ID Kutnar, Klavdija (Avtor)
ID Malnič, Aleksander (Avtor)
ID Marušič, Dragan (Avtor) |
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| Datoteke: |  http://dx.doi.org/10.1016/j.jctb.2007.07.001
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| Jezik: | Angleški jezik |
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| Vrsta gradiva: | Delo ni kategorizirano |
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| Tipologija: | 1.01 - Izvirni znanstveni članek |
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| Organizacija: | IAM - Inštitut Andrej Marušič
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| Opis: | A regular covering projection ▫$\wp : \widetilde{X} \to X$▫ of connected graphs is ▫$G$▫-admissible if ▫$G$▫ lifts along ▫$\wp$▫. Denote by ▫$\tilde{G}$▫ the lifted group, and let CT▫$(\wp)$▫ be the group of covering transformations. The projection is called ▫$G$▫-split whenever the extension ▫{$\mathrm{CT}}(\wp) \to \tilde{G} \to G$▫ 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 ▫$\tilde{G} \cong G$▫ of CT▫$(\wp)$▫ within ▫$\tilde{G}$▫ are transitive on ▫$\widetilde{X}$▫; it is said to be ▫$G$▫-split-sectional whenever for each complement ▫$\tilde{G}$▫ there exists a ▫$\tilde{G}$▫-invariant section of ▫$\wp$▫; 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 ▫$\tilde{G}$▫ is given, and moreover, 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. |
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| Ključne besede: | graph theory, graphs, cubic graphs, symmetric graphs, ▫$s$▫-regular group, regular covering projection |
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| Leto izida: | 2008 |
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| Št. strani: | str. 324-341 |
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| Številčenje: | Vol. 98, no. 2 |
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| PID: | 20.500.12556/RUP-2798  |
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| ISSN: | 0095-8956 |
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| UDK: | 519.17 |
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| COBISS.SI-ID: | 2524887 |
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| Datum objave v RUP: | 15.10.2013 |
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| Število ogledov: | 4270 |
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| Število prenosov: | 35 |
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| Metapodatki: |    |
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