On 2-fold covers of graphs

Feng, Yan-Quan; Kutnar, Klavdija; Malnič, Aleksander; Marušič, Dragan

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Title:	On 2-fold covers of graphs
Authors:	ID Feng, Yan-Quan (Author) ID Kutnar, Klavdija (Author) ID Malnič, Aleksander (Author) ID Marušič, Dragan (Author)
Files:	http://dx.doi.org/10.1016/j.jctb.2007.07.001
Language:	English
Work type:	Not categorized
Typology:	1.01 - Original Scientific Article
Organization:	IAM - Andrej Marušič Institute
Abstract:	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.
Keywords:	graph theory, graphs, cubic graphs, symmetric graphs, $s$ -regular group, regular covering projection
Year of publishing:	2008
Number of pages:	str. 324-341
Numbering:	Vol. 98, no. 2
PID:	20.500.12556/RUP-2798
ISSN:	0095-8956
UDC:	519.17
COBISS.SI-ID:	2524887
Publication date in RUP:	15.10.2013
Views:	4733
Downloads:	36
Metadata:
:	FENG, Yan-Quan, KUTNAR, Klavdija, MALNIČ, Aleksander and MARUŠIČ, Dragan, 2008, On 2-fold covers of graphs. [online]. 2008. Vol. 98, no. 2, p. 324–341. [Accessed 21 April 2025]. Retrieved from: http://dx.doi.org/10.1016/j.jctb.2007.07.001 Copy citation