Locally triangular graphs and rectagraphs with symmetry

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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) such that for each vertex u, the permutation group induced by the vertex stabiliser Gᵤ on the neighbourhood 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 Tₙ. This is because the graph Tₙ, which has vertex set the 2-subsets of {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 the locally 4-homogeneous rectagraphs under some additional structural assumptions. We then use this result to classify the connected locally triangular graphs that are also locally rank 3.