The graphical complexity of direct products of permutation groups

Grech, Mariusz

doi:10.1002/jgt.20504

Cited by 7 publications

(7 citation statements)

References 21 publications

(21 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that the second part of this theorem was also shown in [3,Theorem 5.1] This result, with some exceptions, is also true for particular classes G(k) and DG(k) (for details see [7]). In this paper we consider the converse of the theorem above.…”

Section: Introductionmentioning

confidence: 58%

Decompositions of the automorphism groups of edge-colored graphs into the direct product of permutation groups

Grech

2021

Ars Math. Contemp.

Self Cite

View full text Add to dashboard Cite

In the paper Graphical complexity of products of permutation groups, M. Grech, A. Jeż, and A. Kisielewicz have proved that the direct product of automorphism groups of edgecolored graphs is itself the automorphism groups of an edge-colored graph. In this paper, we study the direct product of two permutation groups such that at least one of them fails to be the automorphism group of an edge-colored graph. We find necessary and sufficient conditions for the direct product to be the automorphism group of an edge-colored graph. The same problem is settled for the edge-colored digraphs.

show abstract

Section: Introductionmentioning

confidence: 58%

Decompositions of the automorphism groups of edge-colored graphs into the direct product of permutation groups

Grech

2021

Ars Math. Contemp.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Moreover, observe that although K 4 and IMR 4 are not coprime, we exclude also the case B 3 × S 4 , since K 4 is coprime with IMR 3 . It remains eight possibilities on A × B.…”

Section: A the Direct Product With Different Componentsmentioning

confidence: 95%

“…In [4], we have continued the study of the direct product and we have improved the result from [6]. We have shown that for k ≥ 3, the direct product of two groups from GR(k) either is in GR(k) or is equal to S 3 2 .…”

mentioning

confidence: 88%

Direct products of automorphism groups of graphs

Grech¹

2009

Journal of Graph Theory

Self Cite

View full text Add to dashboard Cite

Abstract:In this article we study the product action of the direct product of automorphism groups of graphs. We generalize the results of Watkins [J. Combin Theory 11 (1971), 95--104], Nowitz and Watkins [Monatsh. Math. 76 (1972), 168--171] and W. Imrich [Israel J. Math. 11 (1972), 258--264], and we show that except for an infinite family of groups S n ×S n , n ≥ 2 and three other groups D 4 ×S 2 , D 4 ×D 4 and S 4 ×S 2 ×S 2 , the direct product of automorphism groups of two graphs is itself the automorphism group of a graph.

show abstract

“…When comparing the results in [7,8,10,14,25] one may observe that usually formulations of theorems concerning graphical representability are more natural and nicer when the problems are considered for edge-colored graphs rather than for simple graphs. In [13] we provide a relatively simple characterization of those cyclic permutation groups that are automorphism groups of edge-colored graphs.…”

Section: B Mariusz Grechmentioning

confidence: 99%