Locally triangular graphs and rectagraphs with symmetry

TL;DR

This paper classifies certain highly symmetric graphs called locally triangular graphs and rectagraphs, revealing their structure and automorphism groups, advancing understanding of symmetry in graph theory.

Contribution

It provides a classification of locally 4-homogeneous rectagraphs under specific conditions and characterizes connected locally triangular graphs that are locally rank 3.

Findings

Classified locally 4-homogeneous rectagraphs with additional assumptions.

Identified all connected locally triangular graphs that are locally rank 3.

Enhanced understanding of symmetry properties in these graph classes.

Abstract

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 $\Gamma$ is locally rank 3 if there exists $G\leq \mathrm{Aut}(\Gamma)$ such that for each vertex $u$, the permutation group induced by the vertex stabiliser $G_u$ on the neighbourhood $\Gamma(u)$ 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 $T_n$. This is because the graph $T_n$, which has vertex set the $2$-subsets of $\{1,\ldots,n\}$ 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.