Cyclic Permutation Groups that are Automorphism Groups of Graphs

Grech, Mariusz; Kisielewicz, Andrzej

doi:10.1007/s00373-019-02096-1

Cited by 5 publications

(3 citation statements)

References 25 publications

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“…We conclude this section observing that the work of Grech and Kisielewicz [4,5,6], which in spirit is trying to give an explicit description of some classes of permutation groups that are 2-closed, does fit within the subject of group representations on digraphs. In particular, some of the results of Grech and Kisielewicz are interesting in the context of DRRs and their generalisations.…”

Section: Introductionmentioning

confidence: 99%

On Haar digraphical representations of groups

Yan

Spiga

2020

Discrete Mathematics

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A group G admits an n-partite digraphical representation if there exists a regular n-partite digraph Γ such that the automorphism group Aut(Γ) of Γ satisfies the following properties:(1) Aut(Γ) is isomorphic to G, (2) Aut(Γ) acts semiregularly on the vertices of Γ and(3) the orbits of Aut(Γ) on the vertex set of Γ form a partition into n parts giving a structure of n-partite digraph to Γ. In this paper, for every positive integer n, we classify the finite groups admitting an n-partite digraphical representation.

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Section: Introductionmentioning

confidence: 99%

On Haar digraphical representations of groups

Yan

Spiga

2020

Discrete Mathematics

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“…König's problem for this class turned out not so easy as it could seem at the first sight. After some partial results containing errors and wrong proofs ( [24,25] corrected in [11]), the final result has been obtained only recently [15,16]. The full description turns out to consist of seven technical conditions concerning possible lengths of the orbits.…”

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confidence: 99%

“…The proof yields also the result that if a cyclic permutation group is the automorphism group of a colored graph, then it is the automorphism group of a colored graph that uses at most 3 colors. Only then may one consider for which cyclic permutation groups 2 colors suffice, which turns out to have a rather technical solution (see [16]). There are many other results in the area showing that considering edge-colored graphs rather than simple graphs is the right approach (see [21]).…”

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confidence: 99%

Abelian permutation groups with graphical representations

Grech

Kisielewicz

2021

J Algebr Comb

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In this paper we characterize those automorphism groups of colored graphs and digraphs that are abelian as abstract groups. This is done in terms of basic permutation group properties. Using Schur’s classical terminology, what we provide is characterizations of the classes of 2-closed and $$2^*$$ 2 ∗ -closed abelian permutation groups. This is the first characterization concerning these classes since they were defined.

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A conjecture on bipartite graphical regular representations

Yan

Spiga

2020

Discrete Mathematics

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In this paper we are concerned with the classification of the finite groups admitting a bipartite DRR and a bipartite GRR.First, we find a natural obstruction in a finite group for not admitting a bipartite GRR. Then we give a complete classification of the finite groups satisfying this natural obstruction and hence not admitting a bipartite GRR. Based on these results and on some extensive computer computations, we state a conjecture aiming to give a complete classification of the finite groups admitting a bipartite GRR.Next, we prove the existence of bipartite DRRs for most of the finite groups not admitting a bipartite GRR found in this paper. Actually, we prove a much stronger result: we give an asymptotic enumeration of the bipartite DRRs over these groups. Again, based on these results and on some extensive computer computations, we state a conjecture aiming to give a complete classification of the finite groups admitting a bipartite DRR.

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Cyclic Permutation Groups that are Automorphism Groups of Graphs

Cited by 5 publications

References 25 publications

On Haar digraphical representations of groups

On Haar digraphical representations of groups

Abelian permutation groups with graphical representations

A conjecture on bipartite graphical regular representations

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