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On the Saxl graph of a permutation group. (English) Zbl 07167592
Summary: Let $$G$$ be a permutation group on a set $$\Omega$$. A subset of $$\Omega$$ is a base for $$G$$ if its pointwise stabiliser in $$G$$ is trivial. In this paper we introduce and study an associated graph $$\Sigma(G)$$, which we call the Saxl graph of $$G$$. The vertices of $$\Sigma(G)$$ are the points of $$\Omega$$, 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 $$\Sigma(G)$$ 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 $$\Sigma(G)$$ 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 $$G = S_n$$ or $$A_n$$ (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.

##### MSC:
 20B35 Subgroups of symmetric groups 20B05 General theory for finite permutation groups 20B15 Primitive groups 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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