Locally triangular graphs and rectagraphs with symmetry

Locally triangular graphs are known to be halved graphs of bipartite rectagraphs, which are connected triangle-free graphs in which every 2-arc lies in a unique quadrangle. A graph Γ   is locally rank 3 if there exists G⩽Aut(Γ)G⩽Aut(Γ) such that for each vertex u  , the permutation group induced by the vertex stabiliser GuGu on the neighbourhood Γ(u)Γ(u) is transitive of rank 3. One natural place to seek locally rank 3 graphs is among the locally triangular graphs, where every induced neighbourhood graph is isomorphic to a triangular graph TnTn. This is because the graph TnTn, which has vertex set the 2-subsets of {1,…,n}{1,…,n} and edge set the pairs of 2-subsets intersecting at one point, admits a rank 3 group of automorphisms. In this paper, we classify a certain family of rectagraphs for which the permutation group induced by Aut(Γ)u on Γ(u)Γ(u) is 4-homogeneous for some vertex u. We then use this result to classify the connected locally triangular graphs that are also locally rank 3.