r-regular families of graph automorphisms

r-regular families of graph automorphisms

An r-regular family F of permutations on a set V contains, for each pair of vertices u,v∈V, exactly r permutations φ mapping u to v. Earlier, 1-regular families of graph automorphisms were used by Gauyacq to define the quasi-Cayley graphs, a class of vertex-transitive graphs that properly contains the class of Cayley graphs, sharing many of their characteristics, and is properly contained in the class of vertex-transitive graphs. We introduce r-regular families to measure how far a vertex-transitive graph is from being quasi-Cayley. As any automorphism group of a graph Γ=(V,E) acting transitively on V with vertex-stabilizers of order r forms an r-regular family on V, every vertex-transitive graph admits an r-regular family of automorphisms for some r≥1. In general, the smallest r for which such a family exists (which we call the quasi-Cayley deficiency of the graph) might be smaller than the order of the vertex-stabilizer of a smallest vertex-transitive automorphism group of the graph (which we call the Cayley deficiency). We investigate the relations between these two parameters for the class of merged Johnson graphs. We prove the existence of Johnson graphs with arbitrarily large quasi-Cayley deficiency, as well as Johnson graphs for which the difference between their Cayley deficiency and their quasi-Cayley deficiency is arbitrarily large.

97-110

Jajcay, Robert

1f906000-0bb5-4e0b-98f3-4d46fdebafc0

Jones, Gareth A.

fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5

1 June 2019

Jajcay, Robert

1f906000-0bb5-4e0b-98f3-4d46fdebafc0

Jones, Gareth A.

fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5

Jajcay, Robert and Jones, Gareth A.
(2019)
r-regular families of graph automorphisms.
*European Journal of Combinatorics*, 79, .
(doi:10.1016/j.ejc.2018.12.002).

## Abstract

An r-regular family F of permutations on a set V contains, for each pair of vertices u,v∈V, exactly r permutations φ mapping u to v. Earlier, 1-regular families of graph automorphisms were used by Gauyacq to define the quasi-Cayley graphs, a class of vertex-transitive graphs that properly contains the class of Cayley graphs, sharing many of their characteristics, and is properly contained in the class of vertex-transitive graphs. We introduce r-regular families to measure how far a vertex-transitive graph is from being quasi-Cayley. As any automorphism group of a graph Γ=(V,E) acting transitively on V with vertex-stabilizers of order r forms an r-regular family on V, every vertex-transitive graph admits an r-regular family of automorphisms for some r≥1. In general, the smallest r for which such a family exists (which we call the quasi-Cayley deficiency of the graph) might be smaller than the order of the vertex-stabilizer of a smallest vertex-transitive automorphism group of the graph (which we call the Cayley deficiency). We investigate the relations between these two parameters for the class of merged Johnson graphs. We prove the existence of Johnson graphs with arbitrarily large quasi-Cayley deficiency, as well as Johnson graphs for which the difference between their Cayley deficiency and their quasi-Cayley deficiency is arbitrarily large.

Full text not available from this repository.

## More information

Accepted/In Press date: 10 December 2018

e-pub ahead of print date: 14 January 2019

Published date: 1 June 2019

## Identifiers

Local EPrints ID: 428063

URI: https://eprints.soton.ac.uk/id/eprint/428063

ISSN: 0195-6698

PURE UUID: aeee3558-a813-4934-9f18-bd72570a1bb0

## Catalogue record

Date deposited: 07 Feb 2019 17:30

Last modified: 07 Feb 2019 17:30

## Export record

## Altmetrics

## Contributors

Author:
Robert Jajcay

## University divisions

## Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics