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r-regular families of graph automorphisms

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.

0195-6698
97-110
Jajcay, Robert
1f906000-0bb5-4e0b-98f3-4d46fdebafc0
Jones, Gareth A.
fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5
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, 97-110. (doi:10.1016/j.ejc.2018.12.002).

Record type: Article

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.

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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

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