A generalisation of Johnson graphs with an application to triple factorisations

In this paper, we introduce a new generalisation of Johnson graphs. The study of these graphs is linked to the study of intransitive triple factorisations Sym(Ω)=ABA of the (finite) symmetric group, where the subgroups A and B are intransitive subgroups of Sym(Ω). Indeed, we give combinatorial arguments to investigate the conditions under which such factorisations exist. We also use combinatorial arguments to study those conditions for which Sym(Ω) is a Geometric ABA-group, that is to say, Sym(Ω)=ABA, A⊈B, B⊈A and AB∩BA=A∪B.