Homogeneous factorisations of Johnson graphs

Abstract

For a graph Γ, subgroups M < G Aut(Γ), and an edge partition E of Γ, the pair (Γ, E) is a (G, M )-homogeneous factorisation if M is vertex-transitive on Γ and fixes setwise each part of E, while G permutes the parts of E transitively. A classification is given of all homogeneous factorisations of finite Johnson graphs. There are three infinite families and nine sporadic examples.

Primitive decompositions of Johnson graphs

“…(This is an improvement on the proof of [11,Proposition 3.3].) Since G is 3-homogeneous, it is transitive on X.…”

“…If (J (12,4), P) is an M 12 -primitive decomposition then P is given by one of the rows of Table 3. Let H be a maximal subgroup of G such that G {A,B} H < G. The maximal subgroups of G are given in [10, p. 33].…”

“…Construction 6.5. Let (X, B) be the Witt design S (5,6,12). Let F be a linked four, that is a set of three mutually disjoint tetrads (sets of size 4) admitting a refinement into six duads (called duads of F ) such that the union of any three duads coming from any two tetrads is a hexad.…”

“…Let F be a linked four, that is a set of three mutually disjoint tetrads (sets of size 4) admitting a refinement into six duads (called duads of F ) such that the union of any three duads coming from any two tetrads is a hexad. Let (12,3). It turns out (MAGMA calculation [3]) there is exactly one linked four F having A ∪ B as a tetrad and A ∩ B as a duad of F , and so P F is the only part of P containing {A, B}.…”

Abelian Homegeneous Factorisations of Graphs

“…In 2003, Lim and Stringer gave characterizations for homogeneous factorisations of complete digraph and Edge-transitive homogeneous factorisations of complete graphs [2], [3]. In 2004, Cuaresma studied homogeneous factorisations of Johnson graph [4]. In 2007, Giudici, Li, Potocnik, and Praeger accomplished homogeneous factorisations of complete multipartite graphs [5].…”