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Extending earlier results of Godsil and of Dobson and Malniăź on Johnson graphs, we characterise those merged Johnson graphs $$J=J(n,k)_I$$J=J(n,k)I which are Cayley graphs, that is, which are connected and have a group of automorphisms acting regularly on the vertices. We also characterise the merged Johnson graphs which are not Cayley graphs but which have a transitive group of automorphisms with vertex-stabilisers of order 2. Even though these merged Johnson graphs are all vertex-transitive, we show that only relatively few of them are Cayley graphs or have a transitive group of automorphisms with vertex-stabilisers of order 2.

Cayley properties of merged Johnson graphs