Counting some finite-fold coverings of a graph

Kwak, Jin Ho; Lee, Jaeun

doi:10.1007/bf02349964

Cited by 14 publications

(3 citation statements)

References 4 publications

(4 reference statements)

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“…These include: counting isomorphism classes of coverings and, more generally, graph bundles, as considered by Hofmeister [15] and Kwak and Lee [17,18]; constructions of regular maps on surfaces based on covering space techniques due to Archdeacon, Gvozdjak, Nedela, Richter,Širáň,Škoviera and Surowski [1,2,14,28,29,37]; and construction of transitive graphs with a prescribed degree of symmetry, for instance by Du, Malnič, Nedela, Marušič, Scapellato, Seifter, Trofimov and Waller [8,22,23,25,26,34]. Lifting and/or projecting techniques play a prominent role also in the study of imprimitive graphs, cf.…”

Section: Historymentioning

confidence: 99%

Lifting Graph Automorphisms by Voltage Assignments

Malnič

Nedela

Škoviera

2000

European Journal of Combinatorics

152

145

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The problem of lifting graph automorphisms along covering projections and the analysis of lifted groups is considered in a purely combinatorial setting. The main tools employed are: (1) a systematic use of the fundamental groupoid; (2) unification of ordinary, relative and permutation voltage constructions into the concept of a voltage space; (3) various kinds of invariance of voltage spaces relative to automorphism groups; and (4) investigation of geometry of the lifted actions by means of transversals over a localization set. Some applications of these results to regular maps on surfaces are given. Because of certain natural applications and greater generality, graphs are allowed to have semiedges. This requires careful re-examination of the whole subject and at the same time leads to simplification and generalization of several known results.

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Section: Historymentioning

confidence: 99%

Lifting Graph Automorphisms by Voltage Assignments

Malnič

Nedela

Škoviera

2000

European Journal of Combinatorics

152

145

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“…On the other hand, in [4,21] there are descriptions of the topological actions of Z 2 2 . Formula for the number of different type actions of Z 2 2 may be obtained from these descriptions using similar methods to the ones in the works of Alexander Mednykh [20] or Jin Ho Kwak [13,14]. In this section we recall, as a matter of completeness, (1) the description of all possible topological actions of Z 2 2 , as a group of orientation preserving homeomorphisms of a closed orientable surface, and (2) a simple formula to count the different topological classes.…”

Section: Topological Classificationmentioning

confidence: 99%

Schottky Uniformizations of Z22 Actions on Riemann Surfaces

Hidalgo¹

2005

Rev. Mat. Complut.

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“…A large part of algebraic graph theory is devoted to analyzing structural properties of graphs with prescribed degree of symmetry in order to classify, enumerate, construct infinite families, and to produce catalogs of particular classes of interesting graphs up to a certain reasonable size. References are too numerous to be listed here, but see for instance [2,6,8,9,10,11,12,14,16,24,25,26,29,31,36,39,40,42,46,47,52,53,55], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On the split structure of lifted groups

Malnič

Požar

2015

Ars Math. Contemp.

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Let ℘ :X → X be a regular covering projection of connected graphs with the group of covering transformations CT ℘ being abelian. Assuming that a group of automorphisms G ≤ Aut X lifts along ℘ to a groupG ≤ AutX, the problem whether the corresponding exact sequence id → CT ℘ →G → G → id splits is analyzed in detail in terms of a Cayley voltage assignment that reconstructs the projection up to equivalence. In the above combinatorial setting the extension is given only implicitly: neitherG nor the action G → Aut CT ℘ nor a 2-cocycle G × G → CT ℘ , are given. Explicitly constructing the coverX together with CT ℘ andG as permutation groups onX is time and space consuming whenever CT ℘ is large; thus, using the implemented algorithms (for instance, HasComplement in MAGMA) is far from optimal. Instead, we show that the minimal required information about the action and the 2-cocycle can be effectively decoded directly from voltages (without explicitly constructing the cover and the lifted group); one could then use the standard method by reducing the problem to solving a linear system of equations over the integers. However, along these lines we here take a slightly different approach which even does not require any knowledge of cohomology. Time and space complexity are formally analyzed whenever CT ℘ is elementary abelian.

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Counting some finite-fold coverings of a graph

Cited by 14 publications

References 4 publications

Lifting Graph Automorphisms by Voltage Assignments

Lifting Graph Automorphisms by Voltage Assignments

Schottky Uniformizations of Z22 Actions on Riemann Surfaces

On the split structure of lifted groups

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