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On the Saxl graph of a permutation group

Published online by Cambridge University Press:  08 August 2018

TIMOTHY C. BURNESS  and
MICHAEL GIUDICI
TIMOTHY C. BURNESS
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW. e-mail: t.burness@bristol.ac.uk
MICHAEL GIUDICI
Affiliation:
Center for the Mathematics of Symmetry and Computation, Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia. e-mail: michael.giudici@uwa.edu.au
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Abstract

Let G be a permutation group on a set Ω. A subset of Ω is a base for G if its pointwise stabiliser in G is trivial. In this paper we introduce and study an associated graph Σ(G), which we call the Saxl graph of G. The vertices of Σ(G) are the points of Ω, and two vertices are adjacent if they form a base for G. This graph encodes some interesting properties of the permutation group. We investigate the connectivity of Σ(G) for a finite transitive group G, as well as its diameter, Hamiltonicity, clique and independence numbers, and we present several open problems. For instance, we conjecture that if G is a primitive group with a base of size 2, then the diameter of Σ(G) is at most 2. Using a probabilistic approach, we establish the conjecture for some families of almost simple groups. For example, the conjecture holds when G = Sn or An (with n > 12) and the point stabiliser of G is a primitive subgroup. In contrast, we can construct imprimitive groups whose Saxl graph is disconnected with arbitrarily many connected components, or connected with arbitrarily large diameter.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 2 , March 2020 , pp. 219 - 248
Copyright
Copyright © Cambridge Philosophical Society 2018 

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Footnotes

†

Supported by ARC Discovery Project DP160102323.

References

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