Neighbour-transitive codes in Johnson graphs

Abstract

The Johnson graph J(v, k) has, as vertices, the k-subsets of a v-set V and as edges the pairs of k-subsets with intersection of size k − 1. We introduce the notion of a neighbour-transitive code in J(v, k). This is a vertex subset Γ such that the subgroup G of graph automorphisms leaving Γ invariant is transitive on both the set Γ of 'codewords' and also the set of 'neighbours' of Γ, which are the non-codewords joined by an edge to some codeword. We classify all examples where the group G is a subgroup of the symmetric group Sym (V) and is intransitive or imprimitive on the underlying v-set V. In the remaining case where G ≤ Sym (V) and G is primitive on V, we prove that, provided distinct codewords are at distance at least 3, then G is 2-transitive on V. We examine many of the infinite families of finite 2-transitive permutation groups and construct surprisingly rich families of examples of neighbour-transitive codes. A major unresolved case remains.

Sporadic neighbour-transitive codes in Johnson graphs

“…Suppose that Γ is a G-strongly incidence transitive code in J(v, k) with δ(Γ ) ≥ 2, and G ≤ Sym (V) such that G is a 2-transitive permutation group on V, and G is not one of the sporadic 2-transitive groups treated in this paper. We divide such 2-transitive groups into three broad families (see [3, The affine 2-transitive groups are analysed in [12,Section 6] and it is shown in [12, Propositions 6.1 and 6.6] that, for a codeword γ viewed as a subset of V, either (i) γ is an affine subspace or complement of an affine subspace, or (ii) q ∈ {4, 16} and either V is 1-dimensional with γ a Baer subline, or V has dimension at least 2 and γ is a subset of class [0, √ q, q] 1 (that is to say, each affine line meets γ in 0, √ q or q points). For the last case, [ [12] and are remarked there as being an open case.…”

“…(see, for example, [3,Chapter 7.4]) It was shown in [12, Theorem 1.2] that a G-neighbour-transitive code Γ with δ(Γ ) ≥ 3 has the following property, called G-strong incidence transitivity: the group G is transitive on Γ and, for γ ∈ Γ , Gγ is transitive on the set of pairs (u, u ′ ) with u ∈ γ, u ′ ∈ V \ γ. In the case where δ(Γ ) = 2, the same theorem shows that G-strong incidence transitivity is equivalent to G-transitivity on pairs (γ, γ 1 ) with γ ∈ Γ, γ 1 ∈ Γ 1 , a property strictly stronger than G-neighbour transitivity, see [12,Remark 1.5]. The major signifiance of [12, Theorem 1.2] for this paper, however, is its final assertion: namely that, if G is primitive on V, then G-strong incidence transitivity implies that G is 2-transitive on V. Since the finite 2-transitive permutation groups are known explicitly as a consequence of the classification of the finite simple groups (see, for example, [3]), this result offers a way forward to a possible classification of the G-strongly incidence transitive codes in J(v, k).…”

“…Neighbour-transitive codes Γ contained in a Johnson graph J(v, k) were first studied by Liebler and the second author in [12]. All such codes for which the group G := A ∩ Sym (V) does not act primitively on the underlying set V were explicitly described in [12,Theorem 1.1].…”

“…Neighbour-transitive codes Γ contained in a Johnson graph J(v, k) were first studied by Liebler and the second author in [12]. All such codes for which the group G := A ∩ Sym (V) does not act primitively on the underlying set V were explicitly described in [12,Theorem 1.1]. There are two infinite families of examples for which G is intransitive in its action on V. If G is transitive but imprimitive on V, then the classification yields five infinite families of examples together with a recursive construction of such codes.…”

Regularity and planarity of token graphs

“…Thus, the Johnson graph J(n, k) is isomorphic to the k-token graph of the complete graph K n , i.e., J(n, k) F k (K n ). Therefore, results obtained for token graphs also apply for Johnson graphs; Johnson graphs are widely studied due to connections with coding theory, see, e.g., [3,5,6,9,11]. We write u ∼ v whenever u and v are adjacent vertices in G. The edge joining these vertices is denoted by [u, v].…”