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On 2-fold covers of graphs
ID
Feng, Yan-Quan
(
Avtor
),
ID
Kutnar, Klavdija
(
Avtor
),
ID
Malnič, Aleksander
(
Avtor
),
ID
Marušič, Dragan
(
Avtor
)
URL - Predstavitvena datoteka, za dostop obiščite
http://dx.doi.org/10.1016/j.jctb.2007.07.001
Galerija slik
Izvleček
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.
Jezik:
Angleški jezik
Ključne besede:
graph theory
,
graphs
,
cubic graphs
,
symmetric graphs
,
▫$s$▫-regular group
,
regular covering projection
Vrsta gradiva:
Delo ni kategorizirano
Tipologija:
1.01 - Izvirni znanstveni članek
Organizacija:
PEF - Pedagoška fakulteta
Leto izida:
2008
Št. strani:
Str. 324-341
Številčenje:
Vol. 98, no. 2
PID:
20.500.12556/RUL-84774
UDK:
519.17
ISSN pri članku:
0095-8956
COBISS.SI-ID:
2524887
Datum objave v RUL:
09.09.2016
Število ogledov:
1866
Število prenosov:
281
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Objavi na:
Gradivo je del revije
Naslov:
Journal of combinatorial theory
Skrajšan naslov:
J. comb. theory, Ser. B
Založnik:
Elsevier
ISSN:
0095-8956
COBISS.SI-ID:
25721600
Sekundarni jezik
Jezik:
Slovenski jezik
Ključne besede:
teorija grafov
,
grafi
,
kubični grafi
,
simetrični grafi
,
▫$s$▫-regularna grupa
,
regularna krovna projekcija
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