Luca Sabatini

Breaking symmetries of vertex-transitive cubic graphs

A base for a graph is a set of vertices whose pointwise stabiliser in the automorphism group is trivial. In this talk, I will discuss a classification of finite connected vertex-transitive cubic graphs with base size greater than two. Apart from a short list of small highly symmetric graphs, the only examples belong to an explicit infinite family, the split Praeger–Xu graphs. I will first explain this exceptional family and then turn to the main ideas of the proof, which is quite involved and ultimately relies on the Classification of Finite Simple Groups. The arc-transitive case is essentially combinatorial. In the non-arc-transitive case, vertex stabilisers are highly restricted 2-groups, and the problem becomes one of finding conjugate subgroups with trivial intersection. This leads to proving a stronger intersection property for 2-subgroups of almost simple groups and to asymmetric colourings of permutation 2-groups as an ingredient in the induction. This is joint work with Marco Barbieri and Pablo Spiga.