Locally triangular graphs and rectagraphs with symmetry

Abstract

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 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.

Locally triangular graphs and normal quotients of the n-cube

“…There exists a covering g : Q n → Q n for which 0 g = y and e g i = e K i θ −1 for all i n by [1, Lemma 3.1]. Now π K and gθ agree on {0} ∪ Q n (0), so π K = gθ by [1,Lemma 3.2]. Thus π K ϕ = gπ L , and the diagram commutes.…”

“…For K Aut(Q n ), the normaliser N Aut(Qn) (K) acts naturally on V (Q n ) K by (x K ) g := (x g ) K for all x ∈ V Q n and g ∈ N Aut(Qn) (K). For d K 5, in which case (Q n ) K is a rectagraph, every automorphism of (Q n ) K arises in this way by [1,Proposition 3.4] (cf. [7,Lemma 5]).…”

“…Now Aut(Π) = N/K by Theorem 1.4. Since Γ is locally T n and n 5, Aut(Γ) is the setwise stabiliser of V Γ in Aut(Π) by [1,Lemma 4.2]. Recall that V Γ = {x K ∈ V Π : |x| ≡ i mod 2} where i = 0 or 1 by Lemma 4.1(i).…”

“…Various classifications of locally triangular graphs with symmetry exist, including those that are strongly regular [6], 1-homogeneous [5,Theorem 4.4], and locally rank 3 [1]. These characterisations only admit a few families of graphs, but there are many more examples of locally triangular graphs; for instance, any connected component of the distance 2 graph of a coset graph of a binary linear code with minimum distance at least 7 is locally triangular.…”

“…Specifically, a halved graph of a bipartite rectagraph with c 3 = 3 is locally triangular, and conversely, every connected locally triangular graph is a halved graph of a bipartite rectagraph with c 3 = 3 by [4,Proposition 4.3.9]. Rectagraphs were first named by Perkel [9] and have been studied by various authors including [1,3,8]. Bipartite rectagraphs also have links to geometry, for such graphs are the incidence graphs of semibiplanes.…”