Symmetrical 2-extensions of the 3-dimensional grid. I

Abstract

For a positive integer d, a connected graph Γ is a symmetrical 2-extension of the d-dimensional grid Λ^d if there exists a vertex-transitive group G of automorphisms of Γ and its imprimitivity system σ with blocks of size 2 such that there exists an isomorphism φ of the quotient graph Γ/σ onto Λ^d. The tuple (Γ, G, σ, φ) with specified components is called a realization of the symmetrical 2-extension Γ of the grid Λ^d. Two realizations (Γ₁, G₁, σ₁, φ₁) and (Γ₂, G₂, σ₂, φ₂) are called equivalent if there exists an isomorphism of the graph Γ₁ onto Γ₂ which maps σ₁ onto σ₂. V.I. Trofimov proved that, up to equivalence, there are only finitely many realizations of symmetrical 2-extensions of Λ^d for each positive integer d. E.A. Konovalchik and K.V. Kostousov found all, up to equivalence, realizations of symmetrical 2-extensions of the grid Λ². In this work we found all, up to equivalence, realizations (Γ, G, σ, φ) of symmetrical 2-extensions of the grid Λ³ for which only the trivial automorphism of Γ preserves all blocks of σ. Namely we prove that there are 5573 such realizations, and that among corresponding graphs Γ there are 5350 pairwise non-isomorphic.